# A New Decision Method for Public Opinion Crisis with the Intervention of Risk Perception of the Public

Authors: Zhiying Wang; Xiaodi Liu; Shitao Zhang
Journal: Complexity
 (2019)

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## Abstract

Decision-making for selecting response plans problem (SRPP) has been widely concerning to scholars. However, most of the existing studies on this problem are focused on public emergencies, and little attention has been paid to the decision-makers’ urgent need for solving the SRPP in response to public opinion crisis (POC) that may lead to panic buying of materials derived from public emergencies. POC has obvious characteristics of group behaviors that directly resulted from panics and psychological appeals of the public. Therefore, for solving the SRPP in POC, it is necessary to consider the deep-seated cause that result in panics and psychological appeals of the public, i.e., risk perception of the public (RPP). Firstly, the multicase study is employed to describe the SRPP of POC, and thus eight typical cases are chosen to analyze POC and its relevant response measures. Then, the RPP is described with prospect theory through considering the behavioral characteristics and critical sense of the public, the response measures of decision-makers, and the importance and ambiguity of POC. Further, considering the behavioral characteristics of decision-makers and the impact of alternative response plans on the evolution of POC scenarios, a new decision method for solving the SRPP with the intervention of the RPP is proposed by using cumulative prospect theory and a manner of comparing alternatives for each other. Finally, an example is given to illustrate the potential application and effectiveness of the proposed method.


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## Body

## 1. Introduction

In recent years, with the frequent outbreaks of public emergencies (e.g., the “9.11” terrorist attack, hurricane Sandy in the U.S.; the SARS, Wenchuan earthquake, milk products pollution and H1N1 flu in China; the nuclear leakage in Japan; the earthquake in Haiti), relevant emergency management activities have increasingly become the focus of attention of the international community, governments and scholars [1–3]. It is against this background that the International Federation of Red Cross and Red Crescent Societies (IFRC) clearly defines the types of public emergencies, including hurricanes, tornadoes, typhoons, floods, droughts, earthquakes, volcanoes, epidemics, famines, food safety, man-made disasters, population migration, and technological disasters [4]. Public emergencies can not only endanger the human society directly but also cause a series of secondary or derivative events because of their chain reaction. Thus, this kind of indirect harm cannot be ignored. POC is often one of them and refers to the crisis resulted from the spread of emotions, views, and attitudes of the public [5]. With the rapid development of Internet-based new media and we-media technology, the formation of POC becomes faster and broader, which can alter social psychology and affect social stability in a short period of time. For example, during China’s SARS in 2003, the POC about “epidemic of infectious pneumonia” triggered a rush to buy rice vinegar, isatis root, and medical products. During Japan’s nuclear leakage accident in 2011, the POC on “seawater pollution” and “iodine can resist nuclear radiation” caused a rush for iodized salt in China, iodine tablets in the United States, and seaweed in South Korea. In the early stage of POC, if decision-makers can select and initiate the targeted response plan in time with certain standards and methods, it will be helpful to avoid unnecessary losses; thus this study is of great significance.A large number of cases show that POC usually occurs because of public opinion on public emergencies themselves and decision-makers’ response measures (such as the above- mentioned China’s SARS in 2003 and Japan’s nuclear leakage in 2011) after outbreaks of public emergencies. Therefore, similar to public emergencies, POC has gradually become an important research direction in academia, focusing on evolution process modeling [6, 7], evolution path exploration [8], opinion leader identification [9], and influencing factors analysis [10, 11]. Accordingly, as an important means for decision-makers to deal with social risks or public crises, the selection or evaluation methods of alternative response plans have attracted much attention of scholars [12–22]. Amailef and Lu [12] proposed an ontology-based case-based reasoning (OS-CBR) method, which provides a feasible strategy for the evaluation and start-up of decision support systems in emergency situations by constructing a case retrieval process. Wex et al. [13] analyzed the effect of the distribution efficiency of rescue materials on reducing the harm of natural disasters. Then, by weighting the materials according to their importance, a decision support model was designed to minimize the total distribution time, and the starting strategy of the material distribution plan was given. Hämäläinen et al. [14] proposed a risk analysis method based on multi-attribute utility theory for screening the optimal response plan in emergency activities of nuclear accidents. Geldermann et al. [15] proposed a multicriteria decision support method for solving the optimal selection of multiple feasible alternatives in nuclear or radiological accidents. For decision-making of multicriteria and multiperson in unconventional emergencies, Yu and Lai [16] proposed a distance-based group decision making method (GDM). To quickly and effectively relieve the victims of disasters, Chang et al. [17] proposed a greedy search and multiobjective genetic algorithm (GSMOGA) to adjust the allocation of available resources and generate a feasible emergency logistics scheduling scheme. For solving the multicriteria decision-making problem in fuzzy environment, Fu [18] proposed a fuzzy optimization method and applied it to the scheme selection of reservoir flood control. Yang and Xu [19] established an engineering model of the interaction between decision-makers and emergencies by means of sequential games, which was used to generate the optimal relief scheme for emergencies. Liu et al. [20] proposed a fault tree analysis- (FTA-) based evaluation method of response schemes and applied it to the response of H1N1 infectious diseases. Aiming at the interruption risk of supply chain caused by natural disasters, Nejad et al. [21] suggested a mixed integer programming model to study the emergency preparedness of suppliers in the process of improving emergency response capability of the supply chain. He and Zhuang [22] pointed out that the hazard is determined by both preparation and rescue, and thus they studied the allocation of emergency preparedness and rescue costs by constructing a two-stage dynamic programming model. In addition, some scholars believe that considering the behavioral characteristics of decision-makers in the selection of alternatives is helpful to improve the reliability of decision-making [23–26]. Liu et al. [23], Fan et al. [24], and Li et al. [25] considered the behavioral characteristics (loss aversion, reference dependence, diminishing sensitivity, etc.) of decision-makers and the impact of the alternatives on hazards of emergency scenarios and applied prospect theory to study the selection of alternatives. Wang and Li [26] put forward a set description method of behavioral decision-making in emergencies and then studied the selection of alternatives with cumulative prospect theory by considering the evaluation of scenario attributes and the intervention of alternatives on both scenario hazards and scenario evolution.To sum up, the existing studies have made significant contributions to the identification of the evolution law of public emergencies and POC and the selection of alternative response plans. However, it is worth mentioning that most of the abovementioned achievements on the decision method for selecting the optimal response plan from alternatives are based on public emergencies; little attention has been paid to decision-makers’ urgent need for decision methods in response to POC that may lead to panic buying of materials (e.g., during the POC on “seawater pollution” and “iodine can resist nuclear radiation” triggered by Japan’s nuclear leakage in 2011, the Chinese Government adopted a response plan that combines measures such as holding press conferences, punishing lawbreakers, and increasing the supply of related materials). Different from public emergencies, POC generally has clear promoters and media channels, and its evolution process (including appearance, development, climax, decline, and disappearance [27]) is essentially the change process of behavioral relationships between the public (disseminator, recipient, etc.). From the perspective of cognitive psychology, the deep-seated cause for the formation of this change process is the change of RPP [28]. In fact, the change of RPP is synchronized with the change of POC to some extent, because the level of RPP directly determines the intensity of POC, and the intensity of POC directly reflects the level of RPP. So far, the description of RPP has been widely concerned by scholars, and the related studies are mainly based on sociocultural paradigm and psychometric paradigm [29–33], but there is no quantitative model of RPP in the context of POC. Prospect theory and its improved version (e.g., cumulative prospect theory) [34, 35] have attracted much attention in recent years because of their ability to describe some human decision-making behaviors in risk or uncertainty situations (loss aversion, reference dependency, diminishing sensitivity, etc.) [36–38]. In the field of emergency management, prospect theory has been applied to the public level [39, 40] in addition to the decision-makers level [23–26] described above. Campos-Vazquez and Cuilty [39] used prospect theory to measure the impact of student emotions on the risk and loss aversion parameters when manipulated by drug violence and youth unemployment in Mexico City. Wang et al [40] used prospect theory to describe the RPP on the arrival time of emergency materials in the study of optimal scheduling problem of emergency materials under emergencies. Different from these previous studies, this paper will use prospect theory to describe the perceived value of the public on the supply of materials and then deduce the change and formula of RPP after analyzing the behavioral characteristics and critical sense of the public, the response measures of decision-makers, and the importance and ambiguity of POC. On this basis, considering the behavioral characteristics of decision-makers and the impact of alternative response plans on the evolution of POC scenarios, this paper proposes a new decision method for solving the SRPP in response to POC from the perspective of the intervention of RPP by using cumulative prospect theory and a manner of comparing alternatives for each other.The rest of this paper is arranged as follows. Section2 describes the selecting response plans problem (SRPP) in POC in detail with the multicase study, based on which some influencing factors of RPP are analyzed, and a formula of RPP is derived. Then, Section 3 proposes a new decision method for solving the SRPP in POC from the perspective of the intervention of RPP. Further, in Section 4, an example is given to illustrate the potential application and effectiveness of the proposed method. Finally, Section 5 summarizes and highlights the contributions of this paper.


## 2. The Description of SRPP and RPP in POC

The multicase study has become a universal method for construction and testing of theories because it can guarantee the reliability and make up for some shortcomings in data collection, and relevant research objects are usually extracted from sampling [41]. Therefore, for ensuring the universality of the rest of this paper, the multicase study will be used to describe the SRPP in POC, and then a formula of RPP will be proposed. Note that the different types of mass incidents may be induced by POC in different cases, which can lead to differences in the emphasis of the preparation of alternative response plans. Therefore, it is advisable to take the common mass incident of panic buying of materials as the supporting background of this study. For obeying the principle of triangulation [42], this paper adopts the method of data collection on official websites (e.g., the People’s Daily Online and Xinhua Online) and expert discussion and chooses eight typical cases, i.e., SARS, explosion of Jilin petrochemical enterprise, Wenchuan earthquake, milk products pollution and H1N1 flu in China, earthquake in Haiti, nuclear leakage in Japan, and hurricane Sandy in the United States. The criteria we follow for choosing these cases include (1) these cases have triggered POC and panic buying of materials, and which have serious social hazards; (2) these cases are characterized by group and publicity, and thus are highly representative; (3) the evolution process of these cases is easy to review, which helps to ensure the completeness of collected key details; (4) these cases come from different parts of the world and China, helping to ensure the universality of this study. On this basis, the POC resulted from these cases can be taken as the research object of this paper.


### 2.1. Description of SRPP in POC

Through analyzing the POC and the relevant response measures (i.e., the specific content of response plans carried out) of the decision-makers (relevant government departments, enterprises and institutions, etc.) derived from the above eight typical cases, respectively, the results can be sorted out as shown in Table1.Table 1
The POC and its response measures.
Source of POC Introduction of POC Hazard of POC Response measures China’s SARS in 2003 Diffusion of views and attitudes on “epidemic of infectious pneumonia” On December 15th 2002, the same symptoms appeared in medical staff who had treated patients in Guangdong Province of China, and the POC ensued. In late January 2013, which triggered a rush to buy masks, rice vinegar and isatis root. The Guangdong provincial government adopted measures to crack down on price hikes, issue the statement of the epidemic, remove derelict managers and increase the supply of related materials.  China’s explosion of Jilin petrochemical enterprise in 2005 Diffusion of emotions on “water pollution” and “water stoppage rumors” On November 13th 2005, an explosion occurred in the Jilin petrochemical enterprise of PetroChina. On November 21, the POC was ensued and led to residents of Harbin to snap up purified water, mineral water and packaged fresh milk. China’s State Environmental Protection Administration publicized the public concerns about water pollution. Harbin municipal government issued a four-day ban on water supplies due to overhaul of water pipelines.  China’s Wenchuan earthquake in 2008 Diffusion of the panic on “huge casualties and economic losses” On May 12th 2008, an 8.0 earthquake broke out in Wenchuan, Sichuan, China. Due to the damage caused by the earthquake is great, the POC ensued and led to a rush to buy tents, food and drinking water in many parts of Sichuan. National and local seismological bureau of China held press conferences several times to publish information on the disaster, to clarify rumors, to answer doubts and placate emotions of the public.  China’s milk products pollution in 2008 Diffusion of emotions on “trust crisis of milk products made in mainland China” On September 8th 2008, 14 infants were found to have kidney stones. Since then, similar cases have occurred one after another because of milk products pollution. Thus, the POC ensued and resulted in the mainland people rushing to Hong Kong and Taiwan to buy milk products. The AQSIQ repeatedly announced the progress of investigation the incident, and abolished the national inspection exemption system for food industries. Local authorities in China urged related enterprises to recall tainted products.  China’s H1N1 flu in 2009 Diffusion of the fear and panic on “infectious epidemic” On May 10th 2009, the first suspected case of influenza A (H1N1) was detected in Sichuan, China, triggering the POC that led to protective drugs at local drugstores out of stock. Central government of China took measures such as investigation, immigration control and strengthening material reserves. Chengdu municipal government held press conferences to clarify the related information.  Haiti earthquake in 2010 Diffusion of panic on “casualties and food shortages” On January 2th 2010, a 7.3 earthquake broke out in the Caribbean island of Haiti. The tremendous harm triggered the POC and led to the panic buying and robbery of food. The government of Haiti publicized information on the disaster and appeased the public sentiment. WFP and international community distribute food such as high-energy biscuits to the disaster areas.  Japan’s nuclear leakage accident in 2011 Diffusion of views on “seawater pollution” and “iodine can resist nuclear radiation” On March 11th 2011, a 9.0 earthquake broke out in Japan and triggered a nuclear leakage accident. Since then, the POC ensued and triggered mass grabbing of salt in China. Local government in China answered doubts of the public and clarified rumors in time, and the Chinese NDRC issued timely announcements to punishing those who fabricate rumors.  United States’ hurricane Sandy in 2012 Diffusion of emotions on “hurricane attacks, water and power supply cut-off” On October 29th 2012, hurricane Sandy made landfall in the eastern United States. In previous days, the POC spread rapidly and triggered the panic buying of mineral water, batteries and flashlights. The New York government warned vendors to prohibit price hikes, and appealed to the public to keep abreast of the news from the city government and be prepared for evacuation.Through comparing the above cases, we can find that on the one hand the POC in public emergencies usually has the following hazards. First, it aggravates panics and psychological appeals of the public and increases the difficulty of emergency management of leading public emergencies. Second, mass incidents of materials snapping-up are easily induced when panic reaches a certain height and psychological appeals is not satisfied sufficiently. Therefore, it is necessary and urgent to deal with POC by taking the intervention of panics and psychological appeals of the public as a breakthrough point, and to be more important, panics and psychological appeals of the public are actually the reflection of RPP (refers to the level of perception of the public for their lives and safety threatened [43]). On the other hand, after the appearance of POC in above cases, decision-makers (central and local government departments, enterprises, institutions, etc.) have adopted relevant response plans in different degrees. However, as described in the literature review, little attention has been paid to the decision-makers’ urgent need for decision methods in response to POC that may lead to panic buying of materials in the existing studies.In fact, the RPP has an obvious psychological quality, i.e., which is the public’s cognitive and psychological response to threats to some valuable things [44], including both risk assessment and safety education management for disasters [45]. According to this essential meaning of RPP and the accounting method of social welfare for disaster risk management in [46], optimizing behaviors of RPP is consistent with the goal of improving social welfare that decision-makers pay attention to when selecting the optimal response plan. For this reason, the following two problems should be taken into account when a certain scenario of POC that may lead to panic buying of materials appears. One is how to determine a decision method for the SRPP in POC that may lead to panic buying of materials from the perspective of intervention of RPP in time. The other one is how to solve this SRPP and obtain the optimal response plan from alternatives with the determined method. These two problems are significant for decision-makers to face in practice and are key scientific problems worthy of our in-depth study. In view of this, this paper studies how to propose a new decision method and apply it to solve the SRPP in POC that may lead to panic buying of materials (i.e., determine the optimal response plan from alternatives) from the perspective of the intervention of RPP with the consideration of behavioral characteristics (loss aversion, reference dependence, diminishing sensitivity, etc.) of decision-makers. To sum up, the description of the SRPP in POC and its solution with the proposed method in this paper are given by referring to the drawing ideas in [47], as shown in Figure 1.Figure 1
The description of the SRPP in POC and its solution with the proposed method.


### 2.2. Description of RPP in POC

According to the cases studied above, the supply of materials that the public may snap-purchase is also an important decision for decision-makers besides appeasing sentiments of the public and cracking down on illegal behaviors. In fact, the actual supply of materials greatly affects the level of RPP. Specifically, according to the theory of memory manipulation proposed by Sarafidis [48], memory has the effects of proximity, imitation and similarity, and the public will observe the supply of materials at present in the POC-impacted regions (e.g., state, province, city, district, county, community) they concern by referring to that in the past. That is, if the public observe the supply of materials at present is not as good as that in the past, then the RPP increases; otherwise, the RPP does not change. In addition, the application of prospect theory in the decision-making of the public [39, 40] shows that the public usually has the behavioral characteristics of loss aversion and diminishing sensitivity. Under the background of this study, these two kinds of behavioral characteristics can be explained as follows: if the public observe that the current supply of materials is not as good as that of before, they feel “loss”; otherwise, they feel “gain”. Meanwhile, the public generally consider the utility produced from “loss” is larger than that produced from the equivalent “gain”, and their sensitivity to “loss” or “gain” is gradually diminishing [49].Based on the above considerations, it is assumed that the total number of POC-impacted regions the public concern isA, hence the region a∈{1,2,⋯,A}. The total amount of the past period of time that the public observe for supply of materials is T, hence the past period of time t∈{1,2,⋯,T}. Accordingly, the supply of materials in region a during the past period of time t is d^at. Considering the difficulty in obtaining accurate data and the bounded rationality of the public [50, 51], d^at is set to be a fuzzy number [52] and its expected value is dat. Therefore, the average DAT of the past supply of materials in the regions that the public observe is as follows:(1)DAT=1AT∑a=1A∑t=1TdatOn this basis, if it is assumed thatda(T+1) represents the supply of materials in region a during the current time period, then the average DA(T+1) of supply of materials in the regions the public observe during the current time period can similarly be obtained, namely:(2)DAT+1=1A∑a=1AdaT+1Considering the above analysis for behavioral characteristics of the public such as reference dependence, loss aversion, and diminishing sensitivity, this paper uses the value function of the prospect theory [34, 35] to obtain the perceived value v(DA(T+1)) of the public for DA(T+1), that is,(3)vDAT+1=DAT+1-DATα1,DAT+1-DAT≥0;-λ1-DAT+1-DATβ1,DAT+1-DAT<0.where DA(T+1)-DAT≥0 and DA(T+1)-DAT<0 denote the “gain” and “loss” of the public respectively when DAT is taken as the reference point. Correspondingly, α1 and β1 denote the convexity of the perceived value function on the “gain” and “loss” side, respectively, and they depict the diminishing sensitivity of the perceived value of the public. λ1 (λ1>1) describes the loss aversion of the public, that is, for the same amount of the “gain” and “loss”, and the public is more sensitive to the latter.Allport and Postman pointed out that the importance and ambiguity of events would affect the appearance of rumors and proposed a famous rumor formula, namely,R=I×A [53, 54]. Then Chorus considered that critical sense of the public was also an important factor affecting the rumor production and improved the rumor formula to R=I×A/C [55]. Since rumors can be regarded as a product of the evolution of POC, the importance and ambiguity of POC and critical sense of the public also affect the perceived value of the public, in addition to the above-mentioned behavioral characteristics of the public and the supply of materials of decision-makers. On the one hand, the loss aversion of the public usually makes the importance and ambiguity of POC affect their judgment on the current and past situations and easily causes them to deliberately amplify the psychological reference point. On the other hand, [55] points out that the critical sense includes three aspects, namely, knowledge level, wisdom and insight, and moral values, so that the stronger the critical sense of the public is, the greater the cognitive accuracy of the psychological reference point is. Thus, the importance and ambiguity parameter ρ (ρ≥1) of POC and the critical sense parameter θ (θ≥1) of the public are introduced, and formula (3) is improved as follows:(4)vDAT+1,ρ,θ=DAT+1-ρθDATα1,DAT+1-ρθDAT≥0;-λ1-DAT+1-ρθDATβ1,DAT+1-ρθDAT<0.where v(DA(T+1),ρ,θ) denotes the perceived value of the public to DA(T+1) on the premise that the importance and ambiguity of POC is ρ and the critical sense of the public is θ.As can be seen from this,v(DA(T+1),ρ,θ) has a graphical representation as shown in Figure 2. In addition, if r(DA(T+1),ρ,θ) is defined as the RPP for DA(T+1) on the premise that the importance and ambiguity of POC is ρ and the critical sense of the public is θ; then the function curve of r(DA(T+1),ρ,θ) can be obtained by analyzing the function curve of v(DA(T+1),ρ,θ), as shown in Figure 3.Figure 2
Curve of perceived value of the public.Figure 3
Curve of risk perception of the public.In Figure2, if DA(T+1)=(ρ/θ)DAT, then v(DA(T+1),ρ,θ)=0, which denotes that the public believe the supply of materials in the regions at present is roughly equal to that in the past, thus their attitude of hesitation appears and their risk perception to be constant, that is r(DA(T+1),ρ,θ)=r0, as shown in Figure 3. In Figure 2, if DA(T+1)>(ρ/θ)DAT, then v(DA(T+1),ρ,θ)>0, and the growth rate of v(DA(T+1),ρ,θ) decreases with DA(T+1) increases. Which indicates that the public believe the supply of materials in the regions at present is better than that in the past, hence they do not believe the POC and their risk perception decreases, that is, r(DA(T+1),ρ,θ)<r0, as shown in Figure 3. In Figure 2, if DA(T+1)<(ρ/θ)DAT, then v(DA(T+1),ρ,θ)<0, v(DA(T+1),ρ,θ) rapidly decreases and its decreasing rate declines with the decreasing DA(T+1), which indicates that the public believe the supply of materials in the regions at present is not as good as that in the past; hence they believe the POC to some extent and their risk perception increases, that is r(DA(T+1),ρ,θ)>r0, as shown in Figure 3.Furthermore, comparing the function curve ofv(DA(T+1),ρ,θ) with that of r(DA(T+1),ρ,θ), we conclude that the curve of r(DA(T+1),ρ,θ) can be approximately obtained by transforming the curve of v(DA(T+1),ρ,θ) symmetrically along DA(T+1)-axis and then moving up r0 along v(DA(T+1),ρ,θ)-axis. Therefore, the formula of r(DA(T+1),ρ,θ) can be obtained according to formula (4), that is,(5)rDAT+1,ρ,θ=r0-DAT+1-ρθDATα1,DAT+1-ρθDAT≥0;r0+λ1-DAT+1-ρθDATβ1,DAT+1-ρθDAT<0.


## 2.1. Description of SRPP in POC

Through analyzing the POC and the relevant response measures (i.e., the specific content of response plans carried out) of the decision-makers (relevant government departments, enterprises and institutions, etc.) derived from the above eight typical cases, respectively, the results can be sorted out as shown in Table1.Table 1
The POC and its response measures.
Source of POC Introduction of POC Hazard of POC Response measures China’s SARS in 2003 Diffusion of views and attitudes on “epidemic of infectious pneumonia” On December 15th 2002, the same symptoms appeared in medical staff who had treated patients in Guangdong Province of China, and the POC ensued. In late January 2013, which triggered a rush to buy masks, rice vinegar and isatis root. The Guangdong provincial government adopted measures to crack down on price hikes, issue the statement of the epidemic, remove derelict managers and increase the supply of related materials.  China’s explosion of Jilin petrochemical enterprise in 2005 Diffusion of emotions on “water pollution” and “water stoppage rumors” On November 13th 2005, an explosion occurred in the Jilin petrochemical enterprise of PetroChina. On November 21, the POC was ensued and led to residents of Harbin to snap up purified water, mineral water and packaged fresh milk. China’s State Environmental Protection Administration publicized the public concerns about water pollution. Harbin municipal government issued a four-day ban on water supplies due to overhaul of water pipelines.  China’s Wenchuan earthquake in 2008 Diffusion of the panic on “huge casualties and economic losses” On May 12th 2008, an 8.0 earthquake broke out in Wenchuan, Sichuan, China. Due to the damage caused by the earthquake is great, the POC ensued and led to a rush to buy tents, food and drinking water in many parts of Sichuan. National and local seismological bureau of China held press conferences several times to publish information on the disaster, to clarify rumors, to answer doubts and placate emotions of the public.  China’s milk products pollution in 2008 Diffusion of emotions on “trust crisis of milk products made in mainland China” On September 8th 2008, 14 infants were found to have kidney stones. Since then, similar cases have occurred one after another because of milk products pollution. Thus, the POC ensued and resulted in the mainland people rushing to Hong Kong and Taiwan to buy milk products. The AQSIQ repeatedly announced the progress of investigation the incident, and abolished the national inspection exemption system for food industries. Local authorities in China urged related enterprises to recall tainted products.  China’s H1N1 flu in 2009 Diffusion of the fear and panic on “infectious epidemic” On May 10th 2009, the first suspected case of influenza A (H1N1) was detected in Sichuan, China, triggering the POC that led to protective drugs at local drugstores out of stock. Central government of China took measures such as investigation, immigration control and strengthening material reserves. Chengdu municipal government held press conferences to clarify the related information.  Haiti earthquake in 2010 Diffusion of panic on “casualties and food shortages” On January 2th 2010, a 7.3 earthquake broke out in the Caribbean island of Haiti. The tremendous harm triggered the POC and led to the panic buying and robbery of food. The government of Haiti publicized information on the disaster and appeased the public sentiment. WFP and international community distribute food such as high-energy biscuits to the disaster areas.  Japan’s nuclear leakage accident in 2011 Diffusion of views on “seawater pollution” and “iodine can resist nuclear radiation” On March 11th 2011, a 9.0 earthquake broke out in Japan and triggered a nuclear leakage accident. Since then, the POC ensued and triggered mass grabbing of salt in China. Local government in China answered doubts of the public and clarified rumors in time, and the Chinese NDRC issued timely announcements to punishing those who fabricate rumors.  United States’ hurricane Sandy in 2012 Diffusion of emotions on “hurricane attacks, water and power supply cut-off” On October 29th 2012, hurricane Sandy made landfall in the eastern United States. In previous days, the POC spread rapidly and triggered the panic buying of mineral water, batteries and flashlights. The New York government warned vendors to prohibit price hikes, and appealed to the public to keep abreast of the news from the city government and be prepared for evacuation.Through comparing the above cases, we can find that on the one hand the POC in public emergencies usually has the following hazards. First, it aggravates panics and psychological appeals of the public and increases the difficulty of emergency management of leading public emergencies. Second, mass incidents of materials snapping-up are easily induced when panic reaches a certain height and psychological appeals is not satisfied sufficiently. Therefore, it is necessary and urgent to deal with POC by taking the intervention of panics and psychological appeals of the public as a breakthrough point, and to be more important, panics and psychological appeals of the public are actually the reflection of RPP (refers to the level of perception of the public for their lives and safety threatened [43]). On the other hand, after the appearance of POC in above cases, decision-makers (central and local government departments, enterprises, institutions, etc.) have adopted relevant response plans in different degrees. However, as described in the literature review, little attention has been paid to the decision-makers’ urgent need for decision methods in response to POC that may lead to panic buying of materials in the existing studies.In fact, the RPP has an obvious psychological quality, i.e., which is the public’s cognitive and psychological response to threats to some valuable things [44], including both risk assessment and safety education management for disasters [45]. According to this essential meaning of RPP and the accounting method of social welfare for disaster risk management in [46], optimizing behaviors of RPP is consistent with the goal of improving social welfare that decision-makers pay attention to when selecting the optimal response plan. For this reason, the following two problems should be taken into account when a certain scenario of POC that may lead to panic buying of materials appears. One is how to determine a decision method for the SRPP in POC that may lead to panic buying of materials from the perspective of intervention of RPP in time. The other one is how to solve this SRPP and obtain the optimal response plan from alternatives with the determined method. These two problems are significant for decision-makers to face in practice and are key scientific problems worthy of our in-depth study. In view of this, this paper studies how to propose a new decision method and apply it to solve the SRPP in POC that may lead to panic buying of materials (i.e., determine the optimal response plan from alternatives) from the perspective of the intervention of RPP with the consideration of behavioral characteristics (loss aversion, reference dependence, diminishing sensitivity, etc.) of decision-makers. To sum up, the description of the SRPP in POC and its solution with the proposed method in this paper are given by referring to the drawing ideas in [47], as shown in Figure 1.Figure 1
The description of the SRPP in POC and its solution with the proposed method.


## 2.2. Description of RPP in POC

According to the cases studied above, the supply of materials that the public may snap-purchase is also an important decision for decision-makers besides appeasing sentiments of the public and cracking down on illegal behaviors. In fact, the actual supply of materials greatly affects the level of RPP. Specifically, according to the theory of memory manipulation proposed by Sarafidis [48], memory has the effects of proximity, imitation and similarity, and the public will observe the supply of materials at present in the POC-impacted regions (e.g., state, province, city, district, county, community) they concern by referring to that in the past. That is, if the public observe the supply of materials at present is not as good as that in the past, then the RPP increases; otherwise, the RPP does not change. In addition, the application of prospect theory in the decision-making of the public [39, 40] shows that the public usually has the behavioral characteristics of loss aversion and diminishing sensitivity. Under the background of this study, these two kinds of behavioral characteristics can be explained as follows: if the public observe that the current supply of materials is not as good as that of before, they feel “loss”; otherwise, they feel “gain”. Meanwhile, the public generally consider the utility produced from “loss” is larger than that produced from the equivalent “gain”, and their sensitivity to “loss” or “gain” is gradually diminishing [49].Based on the above considerations, it is assumed that the total number of POC-impacted regions the public concern isA, hence the region a∈{1,2,⋯,A}. The total amount of the past period of time that the public observe for supply of materials is T, hence the past period of time t∈{1,2,⋯,T}. Accordingly, the supply of materials in region a during the past period of time t is d^at. Considering the difficulty in obtaining accurate data and the bounded rationality of the public [50, 51], d^at is set to be a fuzzy number [52] and its expected value is dat. Therefore, the average DAT of the past supply of materials in the regions that the public observe is as follows:(1)DAT=1AT∑a=1A∑t=1TdatOn this basis, if it is assumed thatda(T+1) represents the supply of materials in region a during the current time period, then the average DA(T+1) of supply of materials in the regions the public observe during the current time period can similarly be obtained, namely:(2)DAT+1=1A∑a=1AdaT+1Considering the above analysis for behavioral characteristics of the public such as reference dependence, loss aversion, and diminishing sensitivity, this paper uses the value function of the prospect theory [34, 35] to obtain the perceived value v(DA(T+1)) of the public for DA(T+1), that is,(3)vDAT+1=DAT+1-DATα1,DAT+1-DAT≥0;-λ1-DAT+1-DATβ1,DAT+1-DAT<0.where DA(T+1)-DAT≥0 and DA(T+1)-DAT<0 denote the “gain” and “loss” of the public respectively when DAT is taken as the reference point. Correspondingly, α1 and β1 denote the convexity of the perceived value function on the “gain” and “loss” side, respectively, and they depict the diminishing sensitivity of the perceived value of the public. λ1 (λ1>1) describes the loss aversion of the public, that is, for the same amount of the “gain” and “loss”, and the public is more sensitive to the latter.Allport and Postman pointed out that the importance and ambiguity of events would affect the appearance of rumors and proposed a famous rumor formula, namely,R=I×A [53, 54]. Then Chorus considered that critical sense of the public was also an important factor affecting the rumor production and improved the rumor formula to R=I×A/C [55]. Since rumors can be regarded as a product of the evolution of POC, the importance and ambiguity of POC and critical sense of the public also affect the perceived value of the public, in addition to the above-mentioned behavioral characteristics of the public and the supply of materials of decision-makers. On the one hand, the loss aversion of the public usually makes the importance and ambiguity of POC affect their judgment on the current and past situations and easily causes them to deliberately amplify the psychological reference point. On the other hand, [55] points out that the critical sense includes three aspects, namely, knowledge level, wisdom and insight, and moral values, so that the stronger the critical sense of the public is, the greater the cognitive accuracy of the psychological reference point is. Thus, the importance and ambiguity parameter ρ (ρ≥1) of POC and the critical sense parameter θ (θ≥1) of the public are introduced, and formula (3) is improved as follows:(4)vDAT+1,ρ,θ=DAT+1-ρθDATα1,DAT+1-ρθDAT≥0;-λ1-DAT+1-ρθDATβ1,DAT+1-ρθDAT<0.where v(DA(T+1),ρ,θ) denotes the perceived value of the public to DA(T+1) on the premise that the importance and ambiguity of POC is ρ and the critical sense of the public is θ.As can be seen from this,v(DA(T+1),ρ,θ) has a graphical representation as shown in Figure 2. In addition, if r(DA(T+1),ρ,θ) is defined as the RPP for DA(T+1) on the premise that the importance and ambiguity of POC is ρ and the critical sense of the public is θ; then the function curve of r(DA(T+1),ρ,θ) can be obtained by analyzing the function curve of v(DA(T+1),ρ,θ), as shown in Figure 3.Figure 2
Curve of perceived value of the public.Figure 3
Curve of risk perception of the public.In Figure2, if DA(T+1)=(ρ/θ)DAT, then v(DA(T+1),ρ,θ)=0, which denotes that the public believe the supply of materials in the regions at present is roughly equal to that in the past, thus their attitude of hesitation appears and their risk perception to be constant, that is r(DA(T+1),ρ,θ)=r0, as shown in Figure 3. In Figure 2, if DA(T+1)>(ρ/θ)DAT, then v(DA(T+1),ρ,θ)>0, and the growth rate of v(DA(T+1),ρ,θ) decreases with DA(T+1) increases. Which indicates that the public believe the supply of materials in the regions at present is better than that in the past, hence they do not believe the POC and their risk perception decreases, that is, r(DA(T+1),ρ,θ)<r0, as shown in Figure 3. In Figure 2, if DA(T+1)<(ρ/θ)DAT, then v(DA(T+1),ρ,θ)<0, v(DA(T+1),ρ,θ) rapidly decreases and its decreasing rate declines with the decreasing DA(T+1), which indicates that the public believe the supply of materials in the regions at present is not as good as that in the past; hence they believe the POC to some extent and their risk perception increases, that is r(DA(T+1),ρ,θ)>r0, as shown in Figure 3.Furthermore, comparing the function curve ofv(DA(T+1),ρ,θ) with that of r(DA(T+1),ρ,θ), we conclude that the curve of r(DA(T+1),ρ,θ) can be approximately obtained by transforming the curve of v(DA(T+1),ρ,θ) symmetrically along DA(T+1)-axis and then moving up r0 along v(DA(T+1),ρ,θ)-axis. Therefore, the formula of r(DA(T+1),ρ,θ) can be obtained according to formula (4), that is,(5)rDAT+1,ρ,θ=r0-DAT+1-ρθDATα1,DAT+1-ρθDAT≥0;r0+λ1-DAT+1-ρθDATβ1,DAT+1-ρθDAT<0.


## 3. The Proposed Method for Solving the SRPP in POC

As described in formula (5), the behavioral characteristics and critical sense of the public, the response measures (i.e., the supply of materials) of decision-makers, and the importance and ambiguity of POC jointly determine the RPP. For this reason, it is necessary to consider the effect of alternative response plans on these four factors when solving the SRPP in POC from the perspective of the intervention of RPP. Among them, the behavioral characteristics (reference dependence, loss aversion, diminishing sensitivity, etc.) and critical sense are inherent in the public; hence they can be assumed to be unaffected by alternative response plans in a short period of time. The effect of alternatives on the response measures of decision-makers can be reflected in the different response measures in different alternatives. The effect of alternatives on the importance and ambiguity of POC can be reflected by introducing different scenarios of POC and considering the importance of each scenario is different and the evolution of scenarios affected by alternatives.


### 3.1. Symbol Definition

For conducting the research on a new decision method for solving the SRPP in POC from the perspective of the intervention of RPP, we first define the following symbols.S = { S 1 , S 2 , ⋯ , S n } denotes a set of current scenarios that may appear during the evolution of POC. The identification of the current scenario is considered as a part of the response of decision-makers in order to be consistent with the observation of the public for response measures in the current scenario described above.E k = { E 1 k , E 2 k , ⋯ , E l k } denotes a set of subsequent scenarios may be evolved by the current scenario Sk of POC.R = { R 1 , R 2 , ⋯ , R m } denotes a set of alternative response plans.D A ( T + 1 ) R = { D A ( T + 1 ) R 1 , D A ( T + 1 ) R 2 , ⋯ , D A ( T + 1 ) R m } indicates a set of the average of current supply of materials in the regions the public concern under carrying out alternatives, where DA(T+1)Ri represents the average under carrying out the alternative plan Ri.ρ ^ = { ρ ^ 1 , ρ ^ 2 , ⋯ , ρ ^ g } denotes a set of indicators used to evaluate the importance and ambiguity of POC.ρ ^ k = { ρ ^ 1 k , ρ ^ 2 k , ⋯ , ρ ^ g k } denotes a set of values of the set ρ^ in the current scenario Sk, and the total value is ρ-k=∑h=1gρ^hk.P = [ P ( E j k R i ) ] m × l = [ p i j k ] m × l denotes a probability matrix of the new scenario Ejk appears, that is, P(EjkRi) (abbreviated as pijk) indicates the probability that the current scenario Sk evolves into the new scenario Ejk after carrying out the alternative plan Ri.ρ ^ k i j = { ρ ^ 1 k i j , ρ ^ 2 k i j , ⋯ , ρ ^ g k i j } denotes a set of values of the set ρ^ when the current scenario Sk evolves into the new scenario Ejk after carrying out the alternative plan Ri, where ρ^hkij=ρ^hkij(EjkRi) is the value of the h-th indicator, and the total value is ρ-kij=∑h=1gρ^hkij.θ - = { θ - 1 , θ - 2 , ⋯ , θ - A } denotes a set of the critical sense parameter of the public in the POC- impacted regions, where θ-a indicates the critical sense of the public in the region a.i ∈ { 1,2 , ⋯ , m }, j∈{1,2,⋯,l}, k∈{1,2,⋯,n}, h∈{1,2,⋯,g}, a∈{1,2,⋯,A}.


### 3.2. The Proposed Method

In fact, limited to the subjective and objective conditions such as asymmetric information, time pressure and limited rationality of decision-makers, the solution of the SRPP will be inevitably affected by behavioral characteristics (e.g., reference dependence, loss aversion, and diminishing decrease [34, 37]) of decision-makers when a scenario of POC appears. However, different from the application of prospect theory to describe behavioral characteristics of the public above, decision-makers need to take into account the perception probability of the effect of alternative response plans on RPP in POC-impacted regions when solving the SRPP. Therefore, the cumulative prospect theory [35] is introduced to describe the behaviors of decision-makers and to present a new decision method for solving the SRPP in POC. In brief, the specific decision process of this method is as follows.Step 1.
Calculate the prospect value of RPP in each POC-impacted region.According to the above symbol definition and formula (5), when the current scenario Sk evolves into the new scenario Ejk after carrying out the alternative plan Ri, the RPP ra(DA(T+1)Ri,ρ-kij,θ-a) in POC-impacted region a can be described as follows:(6)raDAT+1Ri,ρ-kij,θ-a=r0-DAT+1Ri-ρ-kijθ-aDATα1,DAT+1Ri-ρ-kijθ-aDAT≥0;r0+λ1-DAT+1Ri-ρ-kijθ-aDATβ1,DAT+1Ri-ρ-kijθ-aDAT<0.According to the description of SRPP in Section2.1, the intervention effect of RPP is the basic criterion we follow to judge which alternative plan is the best in alternatives. Therefore, it is necessary to calculate the prospect value of RPP after carrying out an alternative plan. In addition, it is necessary to take the manner of comparing alternatives for each other to avoid the deviation probably caused by setting up the reference point artificially when considering the behavioral characteristics of reference dependence of decision-makers. On such basis, the following formula can be obtained.(7)VraRi1=1nm-1∑k=1n∑i2=1,i2≠i1m∑j1=1l∑j2=1ldraDAT+1Ri1,ρ-ki1j1,θ-a,raDAT+1Ri2,ρ-ki2j2,θ-aPSkPEj1kRi1PEj2kRi2Where V(raRi1) denotes the prospect value of RPP ra in the region a after carrying out the alternative plan Ri1. P(Sk) denotes the probability that the current scenario of POC is Sk. 1/n ensures that there is only one current scenario; 1/(m-1) guarantees that the average of prospect values of RPP can be obtained after comparing Ri1 with others. d(·,·) denotes the value function of cumulative prospect theory and satisfies the following formula:(8)draDAT+1Ri1,ρ-ki1j1,θ-a,raDAT+1Ri2,ρ-ki2j2,θ-a=-λ2raDAT+1Ri1,ρ-ki1j1,θ-a-raDAT+1Ri2,ρ-ki2j2,θ-aβ2,raDAT+1Ri1,ρ-ki1j1,θ-a-raDAT+1Ri2,ρ-ki2j2,θ-a≥0;-raDAT+1Ri1,ρ-ki1j1,θ-a-raDAT+1Ri2,ρ-ki2j2,θ-aα2,raDAT+1Ri1,ρ-ki1j1,θ-a-raDAT+1Ri2,ρ-ki2j2,θ-a<0.where ra(DA(T+1)Ri1,ρ-ki1j1,θ-a)-ra(DA(T+1)Ri2,ρ-ki2j2,θ-a)≥0 and ra(DA(T+1)Ri1,ρ-ki1j1,θ-a)-ra(DA(T+1)Ri2,ρ-ki2j2,θ-a)<0 denote the “loss” and “gain” of decision-makers, respectively, when the reference point of RPP ra(DA(T+1)Ri1,ρ-ki1j1,θ-a) is ra(DA(T+1)Ri2,ρ-ki2j2,θ-a) in the region a. Correspondingly, α2 and β2 denote the convexity of the value function on “gain” and “loss” side, respectively, and both of them depict the behavioral characteristics of diminishing sensitivity of decision-makers. λ2 (λ2>1) describes the behavioral characteristics of loss aversion of decision-makers.From formulae (1), (6), (7), and (8), it can be seen that the calculation of V(raRi1) involves the expected value dat of the fuzzy number d^at. Thus, it is necessary to further explore the relationship between d^at and dat. Without loss of generality, let d^at be a triangular fuzzy number [56] denoted as (d^at1,d^at2,d^at3) (where d^at1, d^at2, and d^at3 are the minimum, the most possible value and the maximum, respectively). On such basis, the following formula can be obtained according to [57]:(9)dat=1-μd^at1+d^at2+μd^at32where 0≤μ≤1 depicts the risk attitude of the public and 0.5<μ<1, μ=0.5, and 0<μ<0.5 denote the risk pursuit, neutrality, and aversion, respectively. The public have the behavioral characteristics of loss aversion when analyzing the RPP above; that is, the public often show a risk aversion attitude in the face of deterministic and risky gains and often show a risk pursuit in the face of deterministic and risky losses according to the prospect theory. Without loss of generality, let the compromise principle be used; thus μ=0.5, dat=(d^at1+2d^at2+d^at3)/4, which is the third indicator of Yager [58].Step 2.
Calculate the decision weight of RPP in each POC-impacted region.To avoid the perceived deviation of decision-makers for risk probability, the cumulative probability function in cumulative prospect theory is introduced to describe the decision weight of RPP. To do this, theV(raRi1),i1∈{1,2,⋯,m} obtained by formula (7) is ranked, namely, V(raR(1))≤V(raR(2))≤⋯≤V(raR(e))≤0≤V(raR(e+1))≤⋯≤V(raR(m-1))≤V(raR(m)), where V(raR(b)), b∈{1,2,⋯,m} indicates that the prospect value of RPP in the region a is ranked at b. Thus, if b≤e, then V(raR(b))≤0; otherwise, V(raR(b))≥0. Based on this, it is assumed that P(raR(b)) denotes the probability that RPP ra has the current prospect value caused by the alternative plan R(b). From the formula (7), it is can be seen that the P(raR(b)) is essentially the sum of the multiplication between the probability P(EjkR(b)) that Sk evolves into Ejk after carrying out the alternative plan R(b) and the probability Paj that RPP takes the current value, that is P(raR(b))=∑k=1n∑j=1lP(Sk)P(EjkR(b))Paj.According to the decision weight function in cumulative prospect theory, the perceived probabilities of decision-makers that the RPPra in the region a takes the current prospect value V(raR(b)) caused by the alternative plan R(b) can be expressed as below:(10)π+raRm=w+PraRmπ-raR1=w-PraR1π+raRb=w+∑i=bmPraRi-w+∑i=b+1mPraRi,b∈e+1,e+2,⋯,m-1π-raRb=w-∑i=1bPraRi-w-∑i=1b-1PraRi,b∈2,3,⋯,ewhere π+(·) and π-(·) denote the perceived probability of decision-makers corresponding to V(raR(b))≥0,b∈{e+1,e+2,⋯,m} and V(raR(b))≤0,b∈{1,2,⋯,e}, respectively. w+(·) and w-(·) are the following nonlinear functions.(11)w+p=pγpγ+1-pγ1/γw-p=pδpδ+1-pδ1/δwhere p denotes the probability. γ and δ are parameters that used to describe the curve degree of w+(·) and w-(·), respectively.Step 3.
Calculate the overall prospect value of RPP in each POC-impacted region.To be convenient for us to compare and discuss the calculated results, it is necessary to ensure that different prospect values of RPP in different regions have the same upper and lower bounds. To do this,V(raR(b)) is standardized in the following manner.(12)V-raRb=VraRb-VminraRiVmaxraRi-VminraRi,i∈1,2,⋯,mwhere V-(raR(b)) is the standardized result of V(raR(b)), obviously V-(raR(b))∈[0,1]; Vmax(raR(i))=max{V(raR(1)),V(raR(2)),⋯,V(raR(m))}; Vmin(raR(i))=min{V(raR(1)),V(raR(2)),⋯,V(raR(m))}. Meanwhile, the positive correlation between V-(raR(b)) and V(raR(b)) can be found.Further, the standardized results ofV(raR(b))≥0,b∈{e+1,e+2,⋯,m} and V(raR(b))≤0,b∈{1,2,⋯,e} are denoted as V-+(raR(b)) and V--(raR(b)), respectively. Both V-+(raR(b)) and V--(raR(b)) can be obtained by formula (12), obviously, V-+(raR(b))∈[0,1],V--(raR(b))∈[0,1].To sum up, it can be seen thatV-+(raR(b)) and V--(raR(b)) correspond to the perceived probabilities π+(raR(b)) and π-(raR(b)) of decision-makers, respectively. Thus, the following formula (13) can be derived by considering the prospect value of RPP in all POC-impacted regions:(13)OVRb=∑a=1Ayπ+raRbV-+raRb+1-yπ-raRbV--raRbWhere OV(R(b)) denotes the overall prospect value of RPP in all POC-impacted regions after carrying out the alternative plan R(b). y is an 0-1 integer variable and satisfies:(14)y=0,b∈1,2,⋯,e;1,b∈e+1,e+2,⋯,m.As mentioned above, for determining the optimal response plan from alternatives when a certain scenario of POC appears, decision-makers should focus on the intervention effect of the alternatives on RPP. Meanwhile, combined with the proposed method for solving the SRPP in POC, we can see that the intervention effect of an alternative plan on RPP is reflected in the overall prospect value of RPP in all POC-impacted regions after carrying out this alternative. Therefore, the principle of determining the optimal response plan is which alternative plan can maximize the value of formula (13); in other words, the larger the OV(R(b)) is, the better the alternative plan R(b) is. On this basis, the ranking of the alternatives can be obtained. In particular, if alternatives Ri1 and Ri2 (i1≠i2) satisfy OV(Ri1)=OV(Ri2), then which is better can be determined further by comparing their expected cost, difficulty in manipulation, and start-up timeliness, etc.. Eventually, based on the ranking of the alternatives, decision-makers can determine the optimal response plan in time according to their empirical data (e.g., the need for related material, financial and human resources) learned from the previous emergency drill.


## 3.1. Symbol Definition

For conducting the research on a new decision method for solving the SRPP in POC from the perspective of the intervention of RPP, we first define the following symbols.S = { S 1 , S 2 , ⋯ , S n } denotes a set of current scenarios that may appear during the evolution of POC. The identification of the current scenario is considered as a part of the response of decision-makers in order to be consistent with the observation of the public for response measures in the current scenario described above.E k = { E 1 k , E 2 k , ⋯ , E l k } denotes a set of subsequent scenarios may be evolved by the current scenario Sk of POC.R = { R 1 , R 2 , ⋯ , R m } denotes a set of alternative response plans.D A ( T + 1 ) R = { D A ( T + 1 ) R 1 , D A ( T + 1 ) R 2 , ⋯ , D A ( T + 1 ) R m } indicates a set of the average of current supply of materials in the regions the public concern under carrying out alternatives, where DA(T+1)Ri represents the average under carrying out the alternative plan Ri.ρ ^ = { ρ ^ 1 , ρ ^ 2 , ⋯ , ρ ^ g } denotes a set of indicators used to evaluate the importance and ambiguity of POC.ρ ^ k = { ρ ^ 1 k , ρ ^ 2 k , ⋯ , ρ ^ g k } denotes a set of values of the set ρ^ in the current scenario Sk, and the total value is ρ-k=∑h=1gρ^hk.P = [ P ( E j k R i ) ] m × l = [ p i j k ] m × l denotes a probability matrix of the new scenario Ejk appears, that is, P(EjkRi) (abbreviated as pijk) indicates the probability that the current scenario Sk evolves into the new scenario Ejk after carrying out the alternative plan Ri.ρ ^ k i j = { ρ ^ 1 k i j , ρ ^ 2 k i j , ⋯ , ρ ^ g k i j } denotes a set of values of the set ρ^ when the current scenario Sk evolves into the new scenario Ejk after carrying out the alternative plan Ri, where ρ^hkij=ρ^hkij(EjkRi) is the value of the h-th indicator, and the total value is ρ-kij=∑h=1gρ^hkij.θ - = { θ - 1 , θ - 2 , ⋯ , θ - A } denotes a set of the critical sense parameter of the public in the POC- impacted regions, where θ-a indicates the critical sense of the public in the region a.i ∈ { 1,2 , ⋯ , m }, j∈{1,2,⋯,l}, k∈{1,2,⋯,n}, h∈{1,2,⋯,g}, a∈{1,2,⋯,A}.


## 3.2. The Proposed Method

In fact, limited to the subjective and objective conditions such as asymmetric information, time pressure and limited rationality of decision-makers, the solution of the SRPP will be inevitably affected by behavioral characteristics (e.g., reference dependence, loss aversion, and diminishing decrease [34, 37]) of decision-makers when a scenario of POC appears. However, different from the application of prospect theory to describe behavioral characteristics of the public above, decision-makers need to take into account the perception probability of the effect of alternative response plans on RPP in POC-impacted regions when solving the SRPP. Therefore, the cumulative prospect theory [35] is introduced to describe the behaviors of decision-makers and to present a new decision method for solving the SRPP in POC. In brief, the specific decision process of this method is as follows.Step 1.
Calculate the prospect value of RPP in each POC-impacted region.According to the above symbol definition and formula (5), when the current scenario Sk evolves into the new scenario Ejk after carrying out the alternative plan Ri, the RPP ra(DA(T+1)Ri,ρ-kij,θ-a) in POC-impacted region a can be described as follows:(6)raDAT+1Ri,ρ-kij,θ-a=r0-DAT+1Ri-ρ-kijθ-aDATα1,DAT+1Ri-ρ-kijθ-aDAT≥0;r0+λ1-DAT+1Ri-ρ-kijθ-aDATβ1,DAT+1Ri-ρ-kijθ-aDAT<0.According to the description of SRPP in Section2.1, the intervention effect of RPP is the basic criterion we follow to judge which alternative plan is the best in alternatives. Therefore, it is necessary to calculate the prospect value of RPP after carrying out an alternative plan. In addition, it is necessary to take the manner of comparing alternatives for each other to avoid the deviation probably caused by setting up the reference point artificially when considering the behavioral characteristics of reference dependence of decision-makers. On such basis, the following formula can be obtained.(7)VraRi1=1nm-1∑k=1n∑i2=1,i2≠i1m∑j1=1l∑j2=1ldraDAT+1Ri1,ρ-ki1j1,θ-a,raDAT+1Ri2,ρ-ki2j2,θ-aPSkPEj1kRi1PEj2kRi2Where V(raRi1) denotes the prospect value of RPP ra in the region a after carrying out the alternative plan Ri1. P(Sk) denotes the probability that the current scenario of POC is Sk. 1/n ensures that there is only one current scenario; 1/(m-1) guarantees that the average of prospect values of RPP can be obtained after comparing Ri1 with others. d(·,·) denotes the value function of cumulative prospect theory and satisfies the following formula:(8)draDAT+1Ri1,ρ-ki1j1,θ-a,raDAT+1Ri2,ρ-ki2j2,θ-a=-λ2raDAT+1Ri1,ρ-ki1j1,θ-a-raDAT+1Ri2,ρ-ki2j2,θ-aβ2,raDAT+1Ri1,ρ-ki1j1,θ-a-raDAT+1Ri2,ρ-ki2j2,θ-a≥0;-raDAT+1Ri1,ρ-ki1j1,θ-a-raDAT+1Ri2,ρ-ki2j2,θ-aα2,raDAT+1Ri1,ρ-ki1j1,θ-a-raDAT+1Ri2,ρ-ki2j2,θ-a<0.where ra(DA(T+1)Ri1,ρ-ki1j1,θ-a)-ra(DA(T+1)Ri2,ρ-ki2j2,θ-a)≥0 and ra(DA(T+1)Ri1,ρ-ki1j1,θ-a)-ra(DA(T+1)Ri2,ρ-ki2j2,θ-a)<0 denote the “loss” and “gain” of decision-makers, respectively, when the reference point of RPP ra(DA(T+1)Ri1,ρ-ki1j1,θ-a) is ra(DA(T+1)Ri2,ρ-ki2j2,θ-a) in the region a. Correspondingly, α2 and β2 denote the convexity of the value function on “gain” and “loss” side, respectively, and both of them depict the behavioral characteristics of diminishing sensitivity of decision-makers. λ2 (λ2>1) describes the behavioral characteristics of loss aversion of decision-makers.From formulae (1), (6), (7), and (8), it can be seen that the calculation of V(raRi1) involves the expected value dat of the fuzzy number d^at. Thus, it is necessary to further explore the relationship between d^at and dat. Without loss of generality, let d^at be a triangular fuzzy number [56] denoted as (d^at1,d^at2,d^at3) (where d^at1, d^at2, and d^at3 are the minimum, the most possible value and the maximum, respectively). On such basis, the following formula can be obtained according to [57]:(9)dat=1-μd^at1+d^at2+μd^at32where 0≤μ≤1 depicts the risk attitude of the public and 0.5<μ<1, μ=0.5, and 0<μ<0.5 denote the risk pursuit, neutrality, and aversion, respectively. The public have the behavioral characteristics of loss aversion when analyzing the RPP above; that is, the public often show a risk aversion attitude in the face of deterministic and risky gains and often show a risk pursuit in the face of deterministic and risky losses according to the prospect theory. Without loss of generality, let the compromise principle be used; thus μ=0.5, dat=(d^at1+2d^at2+d^at3)/4, which is the third indicator of Yager [58].Step 2.
Calculate the decision weight of RPP in each POC-impacted region.To avoid the perceived deviation of decision-makers for risk probability, the cumulative probability function in cumulative prospect theory is introduced to describe the decision weight of RPP. To do this, theV(raRi1),i1∈{1,2,⋯,m} obtained by formula (7) is ranked, namely, V(raR(1))≤V(raR(2))≤⋯≤V(raR(e))≤0≤V(raR(e+1))≤⋯≤V(raR(m-1))≤V(raR(m)), where V(raR(b)), b∈{1,2,⋯,m} indicates that the prospect value of RPP in the region a is ranked at b. Thus, if b≤e, then V(raR(b))≤0; otherwise, V(raR(b))≥0. Based on this, it is assumed that P(raR(b)) denotes the probability that RPP ra has the current prospect value caused by the alternative plan R(b). From the formula (7), it is can be seen that the P(raR(b)) is essentially the sum of the multiplication between the probability P(EjkR(b)) that Sk evolves into Ejk after carrying out the alternative plan R(b) and the probability Paj that RPP takes the current value, that is P(raR(b))=∑k=1n∑j=1lP(Sk)P(EjkR(b))Paj.According to the decision weight function in cumulative prospect theory, the perceived probabilities of decision-makers that the RPPra in the region a takes the current prospect value V(raR(b)) caused by the alternative plan R(b) can be expressed as below:(10)π+raRm=w+PraRmπ-raR1=w-PraR1π+raRb=w+∑i=bmPraRi-w+∑i=b+1mPraRi,b∈e+1,e+2,⋯,m-1π-raRb=w-∑i=1bPraRi-w-∑i=1b-1PraRi,b∈2,3,⋯,ewhere π+(·) and π-(·) denote the perceived probability of decision-makers corresponding to V(raR(b))≥0,b∈{e+1,e+2,⋯,m} and V(raR(b))≤0,b∈{1,2,⋯,e}, respectively. w+(·) and w-(·) are the following nonlinear functions.(11)w+p=pγpγ+1-pγ1/γw-p=pδpδ+1-pδ1/δwhere p denotes the probability. γ and δ are parameters that used to describe the curve degree of w+(·) and w-(·), respectively.Step 3.
Calculate the overall prospect value of RPP in each POC-impacted region.To be convenient for us to compare and discuss the calculated results, it is necessary to ensure that different prospect values of RPP in different regions have the same upper and lower bounds. To do this,V(raR(b)) is standardized in the following manner.(12)V-raRb=VraRb-VminraRiVmaxraRi-VminraRi,i∈1,2,⋯,mwhere V-(raR(b)) is the standardized result of V(raR(b)), obviously V-(raR(b))∈[0,1]; Vmax(raR(i))=max{V(raR(1)),V(raR(2)),⋯,V(raR(m))}; Vmin(raR(i))=min{V(raR(1)),V(raR(2)),⋯,V(raR(m))}. Meanwhile, the positive correlation between V-(raR(b)) and V(raR(b)) can be found.Further, the standardized results ofV(raR(b))≥0,b∈{e+1,e+2,⋯,m} and V(raR(b))≤0,b∈{1,2,⋯,e} are denoted as V-+(raR(b)) and V--(raR(b)), respectively. Both V-+(raR(b)) and V--(raR(b)) can be obtained by formula (12), obviously, V-+(raR(b))∈[0,1],V--(raR(b))∈[0,1].To sum up, it can be seen thatV-+(raR(b)) and V--(raR(b)) correspond to the perceived probabilities π+(raR(b)) and π-(raR(b)) of decision-makers, respectively. Thus, the following formula (13) can be derived by considering the prospect value of RPP in all POC-impacted regions:(13)OVRb=∑a=1Ayπ+raRbV-+raRb+1-yπ-raRbV--raRbWhere OV(R(b)) denotes the overall prospect value of RPP in all POC-impacted regions after carrying out the alternative plan R(b). y is an 0-1 integer variable and satisfies:(14)y=0,b∈1,2,⋯,e;1,b∈e+1,e+2,⋯,m.As mentioned above, for determining the optimal response plan from alternatives when a certain scenario of POC appears, decision-makers should focus on the intervention effect of the alternatives on RPP. Meanwhile, combined with the proposed method for solving the SRPP in POC, we can see that the intervention effect of an alternative plan on RPP is reflected in the overall prospect value of RPP in all POC-impacted regions after carrying out this alternative. Therefore, the principle of determining the optimal response plan is which alternative plan can maximize the value of formula (13); in other words, the larger the OV(R(b)) is, the better the alternative plan R(b) is. On this basis, the ranking of the alternatives can be obtained. In particular, if alternatives Ri1 and Ri2 (i1≠i2) satisfy OV(Ri1)=OV(Ri2), then which is better can be determined further by comparing their expected cost, difficulty in manipulation, and start-up timeliness, etc.. Eventually, based on the ranking of the alternatives, decision-makers can determine the optimal response plan in time according to their empirical data (e.g., the need for related material, financial and human resources) learned from the previous emergency drill.


## 4. An Example

Major epidemics of infectious diseases have occurred frequently in recent years, and they often lead to the POC and the snap-up of related protective drugs, such as SARS in 2003 and H1N1 flu in 2009 described above. Therefore, a theoretical example of the SRPP in POC in a country under the background of this kind of major epidemics of infectious diseases is given to illustrate the potential application and effectiveness of the proposed method. Symptomatic incidents often occur in the early stages of outbreaks of major epidemics of infectious diseases, such as suspected viral infectors appear and they have a long-term fever. Since this concerns the safety of the public, some panic and rumors have been promptly triggered, which might as well be recorded as the current scenarioS1 faced by decision-makers, i.e., S={S1} and P(S1)=1. After estimating the situation of the POC, decision-makers have determined the following three alternatives based on emergency preparedness regulations.R 1: Establish a response team to monitor the development of the POC, and notify the relevant enterprises to prepare for the response.R 2: Launch guidance strategies for the POC based on the response preparation, and require enterprises to ensure the supply of materials the public may snap up.R 3: On the basis of the POC guidance, inform the progress of dealing with the epidemic event in a timely manner and order enterprises to increase the supply of relevant materials.After carrying out the alternative planRi,i∈{1,2,3}, the current scenario S1 will evolve into one of the following new scenarios:E 1 1: Panics of the public are effectively alleviated, rumors are reasonably controlled and the POC is declined or disappeared.E 2 1: The spreading scope of the POC is only concentrated in a small part of the country, and most parts the country are not impacted.E 3 1: The POC has an impact on all parts of the country, leading to the spread of serious rumors and panics.In the current scenarioS1, we set up the total number of regions the public concern A=5, and a∈{1,2,3,4,5}. A day is denoted as the unit of the past period of time t the public observe, and T=30, t∈{1,2,⋯,30}. The supply of materials d^at and its expected value dat calculated by formula (9) are shown in Table 2. Meanwhile, we can obtain DAT=5 by formula (1). The spreading scope ρ^1 is set as the evaluation indicator for the importance and ambiguity of POC, and its value ranges from 1 to 7, where 1 and 7 denote the minimum and maximum respectively, thereby ρ^={ρ^1}. Thus, the values of ρ-1ij can be obtained, as shown in Table 3. Under carrying out different alternative plans, the current supply of materials da(T+1)Ri in the region a is shown in Table 4; hence we can obtain DA(T+1)R1=4, DA(T+1)R2=6, DA(T+1)R3=10 by formula (2). Assuming that the values of critical sense θ-a of the public range from 1 to 7, where 1 and 7 indicate the lowest and highest respectively, here we assume θ-a=a,a∈{1,2,3,4,5}. Considering the completeness of alternative plans and the severity of new scenarios, we can obtain the probability of new scenarios appear by analyzing the related historical data with neural network, Bayesian reasoning, evidence reasoning, etc.; here the data given in [26] is used, as shown in Table 5. For other parameters, it may be appropriate to set up as α1=α2=β1=β2=0.88,λ1=λ2=2.25,γ=0.61,δ=0.69 [34, 35]. Based on this, the value of RPP in each region can be obtained by using formula (6) and setting up r0=10, as shown in Table 6. In summary, the decision steps for solving the SRPP in the POC with the proposed method are as follows.Table 2
The supply of materials during the past period of time the public observe.
region 1 region 2 region 3 region 4 region 5 d ^ a t , ∀ t ∈ { 1 , … , 30 } (4,6,8) (1,3,9) (3,4,9) (5,6,11) (2,3,4)  d a t , ∀ t ∈ { 1 , … , 30 } 6 4 5 7 3  D A T 5Table 3
The importance and ambiguity of POC.
ρ - 1 i j E 1 1 E 2 1 E 3 1  R 1 3 5 7  R 2 2 3 5  R 3 1 2 3Table 4
The supply of materials in the current period of time the public observe.
d a ( T + 1 ) R i region 1 region 2 region 3 region 4 region 5 R 1 4 5 3 6 2  R 2 5 8 6 7 4  R 3 7 9 10 13 11Table 5
The relationship between alternative plans and new scenarios.
P ( E j 1 R i ) E 1 1 E 2 1 E 3 1  R 1 0.1 0.3 0.6  R 2 0.2 0.25 0.55  R 3 0.3 0.2 0.5Table 6
The relationship between alternative plans and RPP.
r a E 1 1 E 2 1 E 3 1  R 1 { 28.561 ,  16.776 ,  12.250 ,  9.705 ,  9.000 } { 42.790 ,  24.794 ,  18.177 ,  1 4.593 ,  12.250 } { 56.194 ,  32.227 ,  23.509 ,  1 8.865 ,  15.916 }  R 2 { 9.621 ,  9.000 ,  7.629 ,  6.989 ,  6.613 } { 25.557 ,  13.215 ,  9.000 ,  7.959 ,  7.371 } { 40.025 ,  21.683 ,  14.742 ,  10.664 ,  9.000 }  R 3 { 5.878 ,  4.111 ,  3.539 ,  3.255 ,  3.086 } { 10.000 ,  5.878 ,  4.691 ,  4.111 ,  3.767 } { 19.274 ,  7.760 ,  5.878 ,  4.984 ,  4.458 }Step 1.
V ( r a R i 1 ) and V-(raRi1),a=1,2,3,4,5;i1=1,2,3 are first obtained with formulae (7), (8), and (12), as shown in Table 7 and Table 8, respectively.Table 7
The prospect value of RPP.
V ( r a R i 1 ) r 1 r 2 r 3 r 4 r 5 R 1 -39.953 -26.594 -20.357 -16.735 -14.043  R 2 -8.674 -4.924 -2.835 -1.741 -1.577  R 3 17.888 11.582 8.380 6.591 5.615Table 8
The standardization for the prospect value of RPP.
V - ( r a R i 1 ) r 1 r 2 r 3 r 4 r 5  R 1 0 0 0 0 0  R 2 0.541 0.568 0.610 0.643 0.634  R 3 1 1 1 1 1Step 2.
The prospect valuesV(raRi1),i1=1,2,3 of RPP in different regions are ranked, respectively, and V(raR1)<V(raR2)<0<V(raR3),a=1,2,3,4,5 can be obtained, which is recorded as V(raR(1))<V(raR(2))<0<V(raR(3)),a=1,2,3,4,5. Based on this, if we set up the probability that the RPP takes the current value in the region a as Paj=0.5, a=1,2,3,4,5, j=1,2,3 when S1 evolves into Ej1 after carrying out the alternative plan R(b), then the value of P(raR(b)) can be obtained in conjunction with Table 5, that is, P(raR(b))=∑j=13P(S1)P(Ej1R(b))Paj=0.5, a=1,2,3,4,5, b=1,2,3. Further, substituting P(raR(b))=0.5 into formulae (10) and (11), we can obtain π-(raR(1))=0.454, π-(raR(2))=0.546, π+(raR(3))=0.421,a=1,2,3,4,5.Step 3.
As shown in Table8, V--(r1R(2))=0.541, V--(r2R(2))=0.568, V--(r3R(2))=0.610, and V--(r4R(2))=0.643,V--(raR(1))=0,V-+(raR(3))=1,a=1,2,3,4,5. Based on this, the overall prospect value of RPP in all POC-impacted regions after carrying out the alternative plan R(b),b∈{1,2,3} can be obtained by formulae (13) and (14), namely, OV(R(1))=0, OV(R(2))=1.636, OV(R(3))=2.105. Therefore, according to the principle of determining the optimal response plan, we can obtain R(3)≻R(2)≻R(1) (where “≻” denotes “better than”), i.e., R3≻R2≻R1.In fact, the calculation results are also consistent with the actual situation of emergency decision-making for POC in the context of such a major epidemic of infectious diseases. For example, during the response to the POC caused by SARS in 2003, it is precisely because the characteristics of SARS virus are not clear and the evolution trend of POC is difficult to be identified effectively in the early stage of SARS outbreaks that some rumors and panics failed to draw sufficient attention from decision-makers, i.e., which accords withR1 or R2. As a result, the POC evolves at an amazing speed and lead to large regions of panic buying of materials (thermometers, masks, rice vinegar, isatis root, etc.). Until the Department of Health of Guangdong Province of China and the CPC Central Committee and State Council one after another issued a variety of strong response measures, i.e., which accords with R3, the evolution of the POC is finally subsided and controlled.


## 5. Conclusions

The multicase study is first employed to describe the SRPP in POC that may lead to panic buying of materials derived from public emergencies, and thus eight typical cases are chosen to analyze the POC and its relevant response measures. Then, the SRPP is defined in detail, that is, how to determine the optimal response plan from the alternatives from the perspective of the intervention of RPP with the consideration of behavioral characteristics of decision-makers. To do this, the RPP is described with prospect theory through considering the behavioral characteristics and critical sense of the public, the response measures of decision-makers, and the importance and ambiguity of POC. Further, considering the behavioral characteristics of decision-makers and the impact of alternative plans on the evolution of scenarios of POC, a new decision method for solving the SRPP with the intervention of the RPP is proposed by using cumulative prospect theory and a manner of comparing alternative response plans for each other. The proposed method shows that the different overall prospect values of the RPP after carrying out different alternative plans can be compared to obtain the ranking of alternatives. Finally, an example of the SRPP in POC in the context of major epidemics of infectious diseases is given to illustrate the potential application and effectiveness of the proposed method.This study can not only enrich the methodology of selecting optimal response plan but also provide some theoretical support for how to restrict the outbreak of secondary incidents in POC such as the panic buying of materials. To sum up, the specific contributions of this study are threefold. First, the RPP is described with prospect theory through considering the behavioral characteristics and critical sense of the public, the response measures of decision-makers, and the importance and ambiguity of POC. Second, a new decision method for solving the SRPP in POC that may lead to panic buying of materials is proposed from the perspective of the intervention of alternatives on RPP. Third, a manner of comparing the alternatives for each other is taken to avoid the deviation probably caused by setting up the reference point artificially, and the impact of the alternatives on the evolution of POC scenarios is considered in the proposed method. The above contributions overcome the shortages that the existing studies neglect the decision-makers’ urgent need for decision methods in response to POC that may lead to panic buying of materials in practice.For the future research, there are two directions to be attached great importance to. One is that the SRPP with time constraint probably needs to be solved further, and thus the proposed method in this paper needs to be extended. The reason is that the response time is probably an important factor affecting the accuracy of the decision-making when determining the optimal plan in response to POC. This direction is motivated by the response to natural disasters and environmental emergencies. For example, Wex et al. [13] point out that response plans should be determined within 30 minutes after the outbreak of natural disasters. In addition, according to “the emergency plan for environmental emergencies in Beijing of China” (the revised edition in 2015), the emergency management office in Beijing stipulates that the municipal environmental protection bureau and the district government must submit the general and detailed situation within 10 minutes and 2 hours respectively after environmental emergencies happen. The other is that the SRPP with dynamic alternative response plans need to be studied further due to the content and the quantity of alternatives are probably dynamic or can be updated with the evolution of POC.



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*Source: 9527218-2019-07-31.xml*