a new decision method for public opinion crisis with the

Research ArticleA New Decision Method for Public Opinion Crisis withthe Intervention of Risk Perception of the Public

Zhiying Wang 12 Xiaodi Liu 3 and Shitao Zhang3

1School of Management Science and Engineering Anhui University of Technology Marsquoanshan Anhui 243032 China2Center for Corporate Governance and Operation Anhui University of Technology Marsquoanshan Anhui 243032 China3School of Mathematics and Physics Anhui University of Technology Marsquoanshan Anhui 243032 China

Correspondence should be addressed to Zhiying Wang zywang87163com

Received 9 May 2019 Revised 4 July 2019 Accepted 14 July 2019 Published 31 July 2019

Guest Editor Raul Banos

Copyright copy 2019 Zhiying Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

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

1 Introduction

In recent years with the frequent outbreaks of public emer-gencies (eg the ldquo911rdquo terrorist attack hurricane Sandy inthe US the SARS Wenchuan earthquake milk productspollution andH1N1 flu in China the nuclear leakage in Japanthe earthquake in Haiti) relevant emergency managementactivities have increasingly become the focus of attentionof the international community governments and scholars[1ndash3] It is against this background that the InternationalFederation of Red Cross and Red Crescent Societies (IFRC)clearly defines the types of public emergencies includinghurricanes tornadoes typhoons floods droughts earth-quakes volcanoes epidemics famines food safety man-made disasters population migration and technologicaldisasters [4] Public emergencies can not only endanger thehuman society directly but also cause a series of secondaryor derivative events because of their chain reaction Thus

this kind of indirect harm cannot be ignored POC is oftenone of them and refers to the crisis resulted from thespread of emotions views and attitudes of the public [5]With the rapid development of Internet-based new mediaand we-media technology the formation of POC becomesfaster and broader which can alter social psychology andaffect social stability in a short period of time For exampleduring Chinarsquos SARS in 2003 the POC about ldquoepidemic ofinfectious pneumoniardquo triggered a rush to buy rice vinegarisatis root and medical products During Japanrsquos nuclearleakage accident in 2011 the POC on ldquoseawater pollutionrdquoand ldquoiodine can resist nuclear radiationrdquo caused a rush foriodized salt in China iodine tablets in the United Statesand seaweed in South Korea In the early stage of POC ifdecision-makers can select and initiate the targeted responseplan in time with certain standards and methods it will behelpful to avoid unnecessary losses thus this study is of greatsignificance

HindawiComplexityVolume 2019 Article ID 9527218 14 pageshttpsdoiorg10115520199527218

2 Complexity

A large number of cases show that POC usually occursbecause of public opinion on public emergencies themselvesand decision-makersrsquo response measures (such as the above-mentioned Chinarsquos SARS in 2003 and Japanrsquos nuclear leakagein 2011) after outbreaks of public emergencies Thereforesimilar to public emergencies POC has gradually becomean important research direction in academia focusing onevolution process modeling [6 7] evolution path exploration[8] opinion leader identification [9] and influencing factorsanalysis [10 11] Accordingly as an important means fordecision-makers to deal with social risks or public crisesthe selection or evaluation methods of alternative responseplans have attracted much attention of scholars [12ndash22]Amailef and Lu [12] proposed an ontology-based case-based reasoning (OS-CBR)method which provides a feasiblestrategy for the evaluation and start-up of decision supportsystems in emergency situations by constructing a caseretrieval process Wex et al [13] analyzed the effect of thedistribution efficiency of rescue materials on reducing theharm of natural disasters Then by weighting the materialsaccording to their importance a decision support model wasdesigned to minimize the total distribution time and thestarting strategy of the material distribution plan was givenHamalainen et al [14] proposed a risk analysis method basedon multi-attribute utility theory for screening the optimalresponse plan in emergency activities of nuclear accidentsGeldermann et al [15] proposed a multicriteria decisionsupport method for solving the optimal selection of multiplefeasible alternatives in nuclear or radiological accidentsFor decision-making of multicriteria and multiperson inunconventional emergencies Yu and Lai [16] proposed adistance-based group decision making method (GDM) Toquickly and effectively relieve the victims of disasters Changet al [17] proposed a greedy search andmultiobjective geneticalgorithm (GSMOGA) to adjust the allocation of availableresources and generate a feasible emergency logistics schedul-ing scheme For solving the multicriteria decision-makingproblem in fuzzy environment Fu [18] proposed a fuzzyoptimization method and applied it to the scheme selectionof reservoir flood control Yang and Xu [19] establishedan engineering model of the interaction between decision-makers and emergencies by means of sequential gameswhich was used to generate the optimal relief scheme foremergencies Liu et al [20] proposed a fault tree analysis-(FTA-) based evaluation method of response schemes andapplied it to the response of H1N1 infectious diseases Aimingat the interruption risk of supply chain caused by naturaldisasters Nejad et al [21] suggested a mixed integer pro-gramming model to study the emergency preparedness ofsuppliers in the process of improving emergency responsecapability of the supply chain He and Zhuang [22] pointedout that the hazard is determined by both preparation andrescue and thus they studied the allocation of emergencypreparedness and rescue costs by constructing a two-stagedynamic programming model In addition some scholarsbelieve that considering the behavioral characteristics ofdecision-makers in the selection of alternatives is helpful toimprove the reliability of decision-making [23ndash26] Liu et al[23] Fan et al [24] and Li et al [25] considered the behavioral

characteristics (loss aversion reference dependence dimin-ishing sensitivity etc) of decision-makers and the impactof the alternatives on hazards of emergency scenarios andapplied prospect theory to study the selection of alternativesWang and Li [26] put forward a set description method ofbehavioral decision-making in emergencies and then studiedthe selection of alternatives with cumulative prospect theoryby considering the evaluation of scenario attributes and theintervention of alternatives on both scenario hazards andscenario evolution

To sum up the existing studies have made significantcontributions to the identification of the evolution law ofpublic emergencies and POC and the selection of alternativeresponse plans However it is worth mentioning that most ofthe abovementioned achievements on the decision methodfor selecting the optimal response plan from alternatives arebased on public emergencies little attention has been paidto decision-makersrsquo urgent need for decision methods inresponse to POC that may lead to panic buying of materials(eg during the POCon ldquoseawater pollutionrdquo and ldquoiodine canresist nuclear radiationrdquo triggered by Japanrsquos nuclear leakagein 2011 the Chinese Government adopted a response planthat combines measures such as holding press conferencespunishing lawbreakers and increasing the supply of relatedmaterials) Different frompublic emergencies POC generallyhas clear promoters and media channels and its evolutionprocess (including appearance development climax declineand disappearance [27]) is essentially the change process ofbehavioral relationships between the public (disseminatorrecipient etc) From the perspective of cognitive psychologythe deep-seated cause for the formation of this change processis the change of RPP [28] In fact the change of RPPis synchronized with the change of POC to some extentbecause the level of RPP directly determines the intensityof POC and the intensity of POC directly reflects the levelof RPP So far the description of RPP has been widelyconcerned by scholars and the related studies are mainlybased on sociocultural paradigmand psychometric paradigm[29ndash33] but there is no quantitative model of RPP in thecontext of POC Prospect theory and its improved version(eg cumulative prospect theory) [34 35] have attractedmuch attention in recent years because of their ability todescribe some human decision-making behaviors in risk oruncertainty situations (loss aversion reference dependencydiminishing sensitivity etc) [36ndash38] In the field of emer-gency management prospect theory has been applied tothe public level [39 40] in addition to the decision-makerslevel [23ndash26] described above Campos-Vazquez and Cuilty[39] used prospect theory to measure the impact of studentemotions on the risk and loss aversion parameters whenmanipulated by drug violence and youth unemploymentin Mexico City Wang et al [40] used prospect theory todescribe the RPP on the arrival time of emergency materialsin the study of optimal scheduling problem of emergencymaterials under emergencies Different from these previousstudies this paper will use prospect theory to describe theperceived value of the public on the supply of materialsand then deduce the change and formula of RPP afteranalyzing the behavioral characteristics and critical sense of

Complexity 3

the public the responsemeasures of decision-makers and theimportance and ambiguity of POCOn this basis consideringthe behavioral characteristics of decision-makers and theimpact of alternative response plans on the evolution of POCscenarios this paper proposes a new decision method forsolving the SRPP in response to POC from the perspectiveof the intervention of RPP by using cumulative prospecttheory and a manner of comparing alternatives for eachother

The rest of this paper is arranged as follows Section 2describes the selecting response plans problem (SRPP) inPOC in detail with the multicase study based on which someinfluencing factors of RPP are analyzed and a formula ofRPP is derived Then Section 3 proposes a new decisionmethod for solving the SRPP in POC from the perspective ofthe intervention of RPP Further in Section 4 an example isgiven to illustrate the potential application and effectivenessof the proposed method Finally Section 5 summarizes andhighlights the contributions of this paper

2 The Description of SRPP and RPP in POC

The multicase study has become a universal method forconstruction and testing of theories because it can guaranteethe reliability and make up for some shortcomings in datacollection and relevant research objects are usually extractedfrom sampling [41] Therefore for ensuring the universalityof the rest of this paper the multicase study will be used todescribe the SRPP in POC and then a formula of RPP willbe proposed Note that the different types of mass incidentsmay be induced by POC in different cases which can lead todifferences in the emphasis of the preparation of alternativeresponse plans Therefore it is advisable to take the commonmass incident of panic buying of materials as the supportingbackground of this study For obeying the principle oftriangulation [42] this paper adopts the method of datacollection on official websites (eg the Peoplersquos Daily Onlineand Xinhua Online) and expert discussion and chooses eighttypical cases ie SARS explosion of Jilin petrochemicalenterprise Wenchuan earthquake milk products pollutionandH1N1 flu inChina earthquake inHaiti nuclear leakage inJapan and hurricane Sandy in the United States The criteriawe follow for choosing these cases include (1) these cases havetriggered POC and panic buying ofmaterials and which haveserious social hazards (2) these cases are characterized bygroup and publicity and thus are highly representative (3)the evolution process of these cases is easy to review whichhelps to ensure the completeness of collected key details(4) these cases come from different parts of the world andChina helping to ensure the universality of this study On thisbasis the POC resulted from these cases can be taken as theresearch object of this paper

21 Description of SRPP in POC Through analyzing thePOC and the relevant response measures (ie the specificcontent of response plans carried out) of the decision-makers (relevant government departments enterprises andinstitutions etc) derived from the above eight typical cases

respectively the results can be sorted out as shown inTable 1

Through comparing the above cases we can find thaton the one hand the POC in public emergencies usuallyhas the following hazards First it aggravates panics andpsychological appeals of the public and increases the difficultyof emergency management of leading public emergenciesSecond mass incidents of materials snapping-up are easilyinduced when panic reaches a certain height and psy-chological appeals is not satisfied sufficiently Therefore itis necessary and urgent to deal with POC by taking theintervention of panics and psychological appeals of the publicas a breakthrough point and to be more important panicsand psychological appeals of the public are actually thereflection of RPP (refers to the level of perception of thepublic for their lives and safety threatened [43]) On theother hand after the appearance of POC in above casesdecision-makers (central and local government departmentsenterprises institutions etc) have adopted relevant responseplans in different degrees However as described in theliterature review little attention has been paid to the decision-makersrsquo urgent need for decisionmethods in response to POCthat may lead to panic buying of materials in the existingstudies

In fact the RPP has an obvious psychological quality iewhich is the publicrsquos cognitive and psychological responseto threats to some valuable things [44] including both riskassessment and safety education management for disasters[45] According to this essential meaning of RPP and theaccounting method of social welfare for disaster riskmanage-ment in [46] optimizing behaviors of RPP is consistent withthe goal of improving social welfare that decision-makers payattention to when selecting the optimal response plan Forthis reason the following two problems should be taken intoaccount when a certain scenario of POC that may lead topanic buying of materials appears One is how to determine adecision method for the SRPP in POC that may lead to panicbuying of materials from the perspective of intervention ofRPP in time The other one is how to solve this SRPP andobtain the optimal response plan from alternatives with thedetermined method These two problems are significant fordecision-makers to face in practice and are key scientificproblems worthy of our in-depth study In view of this thispaper studies how to propose a new decision method andapply it to solve the SRPP in POC that may lead to panicbuying ofmaterials (ie determine the optimal response planfrom alternatives) from the perspective of the intervention ofRPPwith the consideration of behavioral characteristics (lossaversion reference dependence diminishing sensitivity etc)of decision-makers To sumup the description of the SRPP inPOC and its solution with the proposedmethod in this paperare given by referring to the drawing ideas in [47] as shownin Figure 1

22 Description of RPP in POC According to the casesstudied above the supply of materials that the public maysnap-purchase is also an important decision for decision-makers besides appeasing sentiments of the public and

4 Complexity

Table 1 The POC and its response measures

Source of POC Introduction of POC Hazard of POC Response measures

Chinarsquos SARS in 2003

Diffusion of viewsand attitudes onldquoepidemic ofinfectious

pneumoniardquo

On December 15th 2002 thesame symptoms appeared inmedical staff who had treated

patients in Guangdong Provinceof China and the POC ensuedIn late January 2013 which

triggered a rush to buy masksrice vinegar and isatis root

The Guangdong provincialgovernment adopted measures tocrack down on price hikes issuethe statement of the epidemicremove derelict managers andincrease the supply of related

materials

Chinarsquos explosion ofJilin petrochemicalenterprise in 2005

Diffusion of emotionson ldquowater pollutionrdquoand ldquowater stoppage

rumorsrdquo

On November 13th 2005 anexplosion occurred in the Jilinpetrochemical enterprise of

PetroChina On November 21the POC was ensued and led toresidents of Harbin to snap up

purified water mineral water andpackaged fresh milk

Chinarsquos State EnvironmentalProtection Administration

publicized the public concernsabout water pollution Harbinmunicipal government issued afour-day ban on water supplies

due to overhaul of waterpipelines

Chinarsquos Wenchuanearthquake in 2008

Diffusion of the panicon ldquohuge casualtiesand economic lossesrdquo

OnMay 12th 2008 an 80earthquake broke out in

Wenchuan Sichuan China Dueto the damage caused by theearthquake is great the POCensued and led to a rush to buytents food and drinking water in

many parts of Sichuan

National and local seismologicalbureau of China held pressconferences several times topublish information on thedisaster to clarify rumors toanswer doubts and placateemotions of the public

Chinarsquos milk productspollution in 2008

Diffusion of emotionson ldquotrust crisis of

milk products madein mainland Chinardquo

On September 8th 2008 14infants were found to have

kidney stones Since then similarcases have occurred one after

another because of milk productspollution Thus the POC ensuedand resulted in the mainlandpeople rushing to Hong Kong

and Taiwan to buy milk products

The AQSIQ repeatedlyannounced the progress of

investigation the incident andabolished the national inspection

exemption system for foodindustries Local authorities inChina urged related enterprises

to recall tainted products

Chinarsquos H1N1 flu in2009

Diffusion of the fearand panic on

ldquoinfectious epidemicrdquo

On May 10th 2009 the firstsuspected case of influenza A

(H1N1) was detected in SichuanChina triggering the POC thatled to protective drugs at local

drugstores out of stock

Central government of Chinatook measures such as

investigation immigrationcontrol and strengtheningmaterial reserves Chengdu

municipal government held pressconferences to clarify the related

information

Haiti earthquake in2010

Diffusion of panic onldquocasualties and food

shortagesrdquo

On January 2th 2010 a 73earthquake broke out in theCaribbean island of Haiti Thetremendous harm triggered thePOC and led to the panic buying

and robbery of food

The government of Haitipublicized information on thedisaster and appeased the public

sentiment WFP andinternational communitydistribute food such as

high-energy biscuits to thedisaster areas

Japanrsquos nuclearleakage accident in2011

Diffusion of views onldquoseawater pollutionrdquoand ldquoiodine can resistnuclear radiationrdquo

On March 11th 2011 a 90earthquake broke out in Japanand triggered a nuclear leakageaccident Since then the POCensued and triggered massgrabbing of salt in China

Local government in Chinaanswered doubts of the public

and clarified rumors in time andthe Chinese NDRC issued timelyannouncements to punishingthose who fabricate rumors

United Statesrsquohurricane Sandy in2012

Diffusion of emotionson ldquohurricane attacks

water and powersupply cut-offrdquo

On October 29th 2012 hurricaneSandy made landfall in the

eastern United States In previousdays the POC spread rapidly andtriggered the panic buying ofmineral water batteries and

flashlights

The New York governmentwarned vendors to prohibit pricehikes and appealed to the publicto keep abreast of the news from

the city government and beprepared for evacuation

Complexity 5

POC

Data collection for POC

Decision information provided by decision-makers Calculation of RPP

emergencies

Current scenarios of POCAreas impacted by POCPrevious supply of materials ineach AreaImportance and ambiguity of POCCritical sense of the publichelliphellip

Set of alternativesSubsequent scenarios of POCProbability matrix of subsequentscenarios occursBehavioral characteristics of thepublic and decision-makershelliphellip

Perceived value ofthe public for currentsupply of materials

e average of theprecious supply ofmaterials

Determine factorsaffecting perceivedvalue of the public

Deduce the calculationformula of RPP

Calculate the prospect value of RPP

Calculate the decision weight of RPP

Calculate the overall prospect value of RPP

Selection of alternatives from theperspective of the intervention of RPP

Response

Figure 1 The description of the SRPP in POC and its solution with the proposed method

cracking down on illegal behaviors In fact the actual supplyof materials greatly affects the level of RPP Specificallyaccording to the theory of memory manipulation proposedby Sarafidis [48] memory has the effects of proximityimitation and similarity and the public will observe thesupply of materials at present in the POC-impacted regions(eg state province city district county community) theyconcern by referring to that in the past That is if the publicobserve the supply ofmaterials at present is not as good as thatin the past then the RPP increases otherwise the RPP doesnot change In addition the application of prospect theoryin the decision-making of the public [39 40] shows thatthe public usually has the behavioral characteristics of lossaversion and diminishing sensitivity Under the backgroundof this study these two kinds of behavioral characteristicscan be explained as follows if the public observe that thecurrent supply of materials is not as good as that of beforethey feel ldquolossrdquo otherwise they feel ldquogainrdquo Meanwhile thepublic generally consider the utility produced from ldquolossrdquo islarger than that produced from the equivalent ldquogainrdquo andtheir sensitivity to ldquolossrdquo or ldquogainrdquo is gradually diminishing[49]

Based on the above considerations it is assumed that thetotal number of POC-impacted regions the public concern is119860 hence the region 119886 isin 1 2 sdot sdot sdot 119860 The total amount ofthe past period of time that the public observe for supply ofmaterials is 119879 hence the past period of time 119905 isin 1 2 sdot sdot sdot 119879Accordingly the supply of materials in region 119886 during thepast period of time 119905 is 119889119886119905 Considering the difficulty inobtaining accurate data and the bounded rationality of thepublic [50 51] 119889119886119905 is set to be a fuzzy number [52] and itsexpected value is 119889119886119905 Therefore the average 119863119860119879 of the past

supply of materials in the regions that the public observe is asfollows

119863119860119879 = 1119860119879119860sum119886=1

119879sum119905=1

119889119886119905 (1)

On this basis if it is assumed that 119889119886(119879+1) representsthe supply of materials in region 119886 during the current timeperiod then the average119863119860(119879+1) of supply of materials in theregions the public observe during the current time period cansimilarly be obtained namely

119863119860(119879+1) = 1119860119860sum119886=1

119889119886(119879+1) (2)

Considering the above analysis for behavioral charac-teristics of the public such as reference dependence lossaversion and diminishing sensitivity this paper uses thevalue function of the prospect theory [34 35] to obtain theperceived value V(119863119860(119879+1)) of the public for 119863119860(119879+1) thatis

V (119863119860(119879+1))

= (119863119860(119879+1) minus 119863119860119879)1205721 119863119860(119879+1) minus 119863119860119879 ge 0minus1205821 [minus (119863119860(119879+1) minus 119863119860119879)]1205731 119863119860(119879+1) minus 119863119860119879 lt 0

(3)

where119863119860(119879+1) minus 119863119860119879 ge 0 and 119863119860(119879+1) minus 119863119860119879 lt 0 denote theldquogainrdquo and ldquolossrdquo of the public respectively when119863119860119879 is taken

6 Complexity

(DA(T+1) )

DA(T+1)

DAT

Figure 2 Curve of perceived value of the public

DA(T+1)

DAT

r(DA(T+1) )

r0

Figure 3 Curve of risk perception of the public

as the reference point Correspondingly 1205721 and 1205731 denotethe convexity of the perceived value function on the ldquogainrdquoand ldquolossrdquo side respectively and they depict the diminishingsensitivity of the perceived value of the public 1205821 (1205821 gt1) describes the loss aversion of the public that is for thesame amount of the ldquogainrdquo and ldquolossrdquo and the public is moresensitive to the latter

Allport and Postman pointed out that the importance andambiguity of events would affect the appearance of rumorsand proposed a famous rumor formula namely 119877 = 119868 times119860 [53 54] Then Chorus considered that critical sense ofthe public was also an important factor affecting the rumorproduction and improved the rumor formula to 119877 = (119868 times119860)119862 [55] Since rumors can be regarded as a product of theevolution of POC the importance and ambiguity of POC andcritical sense of the public also affect the perceived value ofthe public in addition to the above-mentioned behavioralcharacteristics of the public and the supply of materials ofdecision-makers On the one hand the loss aversion of thepublic usually makes the importance and ambiguity of POCaffect their judgment on the current and past situations andeasily causes them to deliberately amplify the psychologicalreference point On the other hand [55] points out thatthe critical sense includes three aspects namely knowledgelevel wisdom and insight and moral values so that thestronger the critical sense of the public is the greater thecognitive accuracy of the psychological reference point isThus the importance and ambiguity parameter 120588 (120588 ge 1)of POC and the critical sense parameter 120579 (120579 ge 1) of

the public are introduced and formula (3) is improved asfollowsV (119863119860(119879+1) 120588 120579)

= (119863119860(119879+1) minus 120588120579119863119860119879)

1205721 119863119860(119879+1) minus 120588120579119863119860119879 ge 0minus1205821 [minus (119863119860(119879+1) minus 120588120579119863119860119879)]

1205731 119863119860(119879+1) minus 120588120579119863119860119879 lt 0(4)

where V(119863119860(119879+1) 120588 120579) denotes the perceived value of thepublic to 119863119860(119879+1) on the premise that the importance andambiguity of POC is 120588 and the critical sense of the public is 120579

As can be seen from this V(119863119860(119879+1) 120588 120579) has a graph-ical representation as shown in Figure 2 In addition if119903(119863119860(119879+1) 120588 120579) is defined as the RPP for 119863119860(119879+1) on thepremise that the importance and ambiguity of POC is 120588 andthe critical sense of the public is 120579 then the function curveof 119903(119863119860(119879+1) 120588 120579) can be obtained by analyzing the functioncurve of V(119863119860(119879+1) 120588 120579) as shown in Figure 3

In Figure 2 if 119863119860(119879+1) = (120588120579)119863119860119879 then V(119863119860(119879+1) 120588 120579)= 0 which denotes that the public believe the supply ofmaterials in the regions at present is roughly equal to thatin the past thus their attitude of hesitation appears and theirrisk perception to be constant that is 119903(119863119860(119879+1) 120588 120579) = 1199030 asshown in Figure 3 In Figure 2 if 119863119860(119879+1) gt (120588120579)119863119860119879 thenV(119863119860(119879+1) 120588 120579) gt 0 and the growth rate of V(119863119860(119879+1) 120588 120579)decreases with 119863119860(119879+1) increases Which indicates that thepublic believe the supply of materials in the regions atpresent is better than that in the past hence they donot believe the POC and their risk perception decreases

Complexity 7

that is 119903(119863119860(119879+1) 120588 120579) lt 1199030 as shown in Figure 3 InFigure 2 if 119863119860(119879+1) lt (120588120579)119863119860119879 then V(119863119860(119879+1) 120588 120579) lt0 V(119863119860(119879+1) 120588 120579) rapidly decreases and its decreasing ratedeclines with the decreasing119863119860(119879+1) which indicates that thepublic believe the supply of materials in the regions at presentis not as good as that in the past hence they believe the POCto some extent and their risk perception increases that is119903(119863119860(119879+1) 120588 120579) gt 1199030 as shown in Figure 3

Furthermore comparing the function curve ofV(119863119860(119879+1) 120588 120579) with that of 119903(119863119860(119879+1) 120588 120579) we concludethat the curve of 119903(119863119860(119879+1) 120588 120579) can be approximatelyobtained by transforming the curve of V(119863119860(119879+1) 120588 120579)symmetrically along 119863119860(119879+1)-axis and then moving up1199030 along V(119863119860(119879+1) 120588 120579)-axis Therefore the formula of119903(119863119860(119879+1) 120588 120579) can be obtained according to formula (4)that is

119903 (119863119860(119879+1) 120588 120579)

= 1199030 minus (119863119860(119879+1) minus 120588120579119863119860119879)

1205721 119863119860(119879+1) minus 120588120579119863119860119879 ge 01199030 + 1205821 [minus(119863119860(119879+1) minus 120588120579119863119860119879)]

1205731 119863119860(119879+1) minus 120588120579119863119860119879 lt 0(5)

3 The Proposed Method for Solvingthe SRPP in POC

As described in formula (5) the behavioral characteristicsand critical sense of the public the response measures(ie the supply of materials) of decision-makers and theimportance and ambiguity of POC jointly determine theRPP For this reason it is necessary to consider the effect ofalternative response plans on these four factors when solvingthe SRPP in POC from the perspective of the intervention ofRPP Among them the behavioral characteristics (referencedependence loss aversion diminishing sensitivity etc) andcritical sense are inherent in the public hence they can beassumed to be unaffected by alternative response plans ina short period of time The effect of alternatives on theresponse measures of decision-makers can be reflected inthe different response measures in different alternatives Theeffect of alternatives on the importance and ambiguity ofPOC can be reflected by introducing different scenarios ofPOC and considering the importance of each scenario isdifferent and the evolution of scenarios affected by alterna-tives

31 Symbol Definition For conducting the research on a newdecision method for solving the SRPP in POC from theperspective of the intervention of RPP we first define thefollowing symbols119878 = 1198781 1198782 sdot sdot sdot 119878119899 denotes a set of current scenarios thatmay appear during the evolution of POC The identificationof the current scenario is considered as a part of theresponse of decision-makers in order to be consistentwith theobservation of the public for responsemeasures in the currentscenario described above

119864119896 = 1198641198961 1198641198962 sdot sdot sdot 119864119896119897 denotes a set of subsequentscenarios may be evolved by the current scenario 119878119896 of POC119877 = 1198771 1198772 sdot sdot sdot 119877119898 denotes a set of alternative responseplans119863119877119860(119879+1) = 1198631198771

119860(119879+1) 1198631198772119860(119879+1)

sdot sdot sdot 119863119877119898119860(119879+1)

indicates aset of the average of current supply of materials in theregions the public concern under carrying out alternativeswhere 119863119877119894

119860(119879+1)represents the average under carrying out the

alternative plan 119877119894120588 = 1205881 1205882 sdot sdot sdot 120588119892 denotes a set of indicators used toevaluate the importance and ambiguity of POC120588119896 = 1205881198961 1205881198962 sdot sdot sdot 120588119896119892 denotes a set of values of the set 120588 inthe current scenario 119878119896 and the total value is 120588119896 = sum119892

ℎ=1120588119896ℎ 119875 = [119875(119864119896119895 |119877119894)]119898times119897 = [119901119896119894119895]119898times119897 denotes a probability matrix

of the new scenario 119864119896119895 appears that is 119875(119864119896119895 |119877119894) (abbreviatedas 119901119896119894119895) indicates the probability that the current scenario119878119896 evolves into the new scenario 119864119896119895 after carrying out thealternative plan 119877119894120588119896119894119895 = 1205881198961198941198951 1205881198961198941198952 sdot sdot sdot 120588119896119894119895119892 denotes a set of values of the set120588 when the current scenario 119878119896 evolves into the new scenario119864119896119895 after carrying out the alternative plan 119877119894 where 120588119896119894119895ℎ =120588119896119894119895ℎ|(119864119896119895 |119877119894) is the value of the h-th indicator and the total

value is 120588119896119894119895 = sum119892ℎ=1

120588119896119894119895ℎ

120579 = 1205791 1205792 sdot sdot sdot 120579119860 denotes a set of the critical senseparameter of the public in the POC- impacted regions where120579119886 indicates the critical sense of the public in the region 119886119894 isin 1 2 sdot sdot sdot 119898 119895 isin 1 2 sdot sdot sdot 119897 119896 isin 1 2 sdot sdot sdot 119899 ℎ isin1 2 sdot sdot sdot 119892 119886 isin 1 2 sdot sdot sdot 11986032 The Proposed Method In fact limited to the subjectiveand objective conditions such as asymmetric informationtime pressure and limited rationality of decision-makersthe solution of the SRPP will be inevitably affected bybehavioral characteristics (eg reference dependence lossaversion and diminishing decrease [34 37]) of decision-makers when a scenario of POC appears However differentfrom the application of prospect theory to describe behavioralcharacteristics of the public above decision-makers need totake into account the perception probability of the effectof alternative response plans on RPP in POC-impactedregions when solving the SRPP Therefore the cumulativeprospect theory [35] is introduced to describe the behaviorsof decision-makers and to present a new decision methodfor solving the SRPP in POC In brief the specific decisionprocess 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 119878119896 evolves into the newscenario 119864119896119895 after carrying out the alternative plan 119877119894 theRPP 119903119886(119863119877119894119860(119879+1) 120588119896119894119895 120579119886) in POC-impacted region 119886 can bedescribed as follows

8 Complexity

119903119886 (119863119877119894119860(119879+1) 120588119896119894119895 120579119886) =

1199030 minus (119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879)

1205721 119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879 ge 0

1199030 + 1205821 [minus(119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879)]

1205731 119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879 lt 0

(6)

According to the description of SRPP in Section 21 theintervention effect of RPP is the basic criterion we follow tojudge which alternative plan is the best in alternatives There-fore it is necessary to calculate the prospect value of RPP aftercarrying out an alternative plan In addition it is necessary

to take the manner of comparing alternatives for each otherto avoid the deviation probably caused by setting up thereference point artificially when considering the behavioralcharacteristics of reference dependence of decision-makersOn such basis the following formula can be obtained

119881(119903119886 100381610038161003816100381610038161198771198941 ) = 1119899 (119898 minus 1)sdot 119899sum119896=1

119898sum1198942=11198942 =1198941

119897sum1198951=1

119897sum1198952=1

119889 [119903119886 (1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) 119903119886 (1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886)]119875 (119878119896) 119875 (1198641198961198951 100381610038161003816100381610038161198771198941 ) 119875 (1198641198961198952 100381610038161003816100381610038161198771198942 )(7)

Where 119881(119903119886|1198771198941) denotes the prospect value of RPP 119903119886 in theregion 119886 after carrying out the alternative plan 1198771198941 119875(119878119896)denotes the probability that the current scenario of POC is 1198781198961119899 ensures that there is only one current scenario 1(119898minus 1)

guarantees that the average of prospect values of RPP can beobtained after comparing 1198771198941 with others 119889(sdot sdot) denotes thevalue function of cumulative prospect theory and satisfies thefollowing formula

119889 [119903119886 (1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) 119903119886 (1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886)]

= minus1205822 [119903119886 (1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) minus 119903119886 (1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886)]1205732 119903119886 (1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) minus 119903119886 (1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886) ge 0minus [119903119886 (1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) minus 119903119886 (1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886)]1205722 119903119886 (1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) minus 119903119886 (1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886) lt 0

(8)

where 119903119886(1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) minus 119903119886(119863R1198942119860(119879+1)

12058811989611989421198952 120579119886) ge 0and 119903119886(1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) minus 119903119886(1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886) lt 0denote the ldquolossrdquo and ldquogainrdquo of decision-makers respectivelywhen the reference point of RPP 119903119886(1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) is119903119886(1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886) in the region 119886 Correspondingly1205722 and1205732 denote the convexity of the value function on ldquogainrdquo andldquolossrdquo side respectively and both of them depict the behav-ioral characteristics of diminishing sensitivity of decision-makers 1205822 (1205822 gt 1) describes the behavioral characteristicsof loss aversion of decision-makers

From formulae (1) (6) (7) and (8) it can be seen thatthe calculation of 119881(119903119886|1198771198941) involves the expected value 119889119886119905of the fuzzy number 119889119886119905 Thus it is necessary to furtherexplore the relationship between 119889119886119905 and 119889119886119905 Without loss ofgenerality let 119889119886119905 be a triangular fuzzy number [56] denotedas (1198891119886119905 1198892119886119905 1198893119886119905) (where 1198891119886119905 1198892119886119905 and 1198893119886119905 are the minimumthe most possible value and the maximum respectively) Onsuch basis the following formula can be obtained accordingto [57]

119889119886119905 = (1 minus 120583) 1198891119886119905 + 1198892119886119905 + 12058311988931198861199052 (9)

where 0 le 120583 le 1 depicts the risk attitude of the public and05 lt 120583 lt 1 120583 = 05 and 0 lt 120583 lt 05 denote the riskpursuit neutrality and aversion respectivelyThe public havethe behavioral characteristics of loss aversion when analyzingthe RPP above that is the public often show a risk aversionattitude in the face of deterministic and risky gains and oftenshow a risk pursuit in the face of deterministic and risky lossesaccording to the prospect theory Without loss of generalitylet the compromise principle be used thus 120583 = 05 119889119886119905 =(1198891119886119905 +21198892119886119905 +1198893119886119905)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-makersfor risk probability the cumulative probability function incumulative prospect theory is introduced to describe thedecision weight of RPP To do this the 119881(119903119886|1198771198941) 1198941 isin1 2 sdot sdot sdot 119898 obtained by formula (7) is ranked namely

Complexity 9

119881(119903119886|119877(1)) le 119881(119903119886|119877(2)) le sdot sdot sdot le 119881(119903119886|119877(119890)) le 0 le119881(119903119886|119877(119890+1)) le sdot sdot sdot le 119881(119903119886|119877(119898minus1)) le 119881(119903119886|119877(119898)) where119881(119903119886|119877(119887)) 119887 isin 1 2 sdot sdot sdot 119898 indicates that the prospect valueof RPP in the region 119886 is ranked at 119887 Thus if 119887 le 119890then 119881(119903119886|119877(119887)) le 0 otherwise 119881(119903119886|119877(119887)) ge 0 Based onthis it is assumed that 119875(119903119886|119877(119887)) denotes the probabilitythat RPP 119903119886 has the current prospect value caused by thealternative plan 119877(119887) From the formula (7) it is can be seenthat the 119875(119903119886|119877(119887)) is essentially the sum of the multiplicationbetween the probability 119875(119864119896119895 |119877(119887)) that 119878119896 evolves into 119864119896119895after carrying out the alternative plan 119877(119887) and the probability119875119895119886 that RPP takes the current value that is 119875(119903119886|119877(119887)) =sum119899119896=1sum119897119895=1 119875(119878119896)119875(119864119896119895 |119877(119887))119875119895119886

According to the decision weight function in cumulativeprospect theory the perceived probabilities of decision-makers that the RPP 119903119886 in the region 119886 takes the currentprospect value 119881(119903119886|119877(119887)) caused by the alternative plan 119877(119887)can be expressed as below

120587+ (119903119886 1003816100381610038161003816119877(119898) ) = 119908+ [119875 (119903119886 1003816100381610038161003816119877(119898) )]120587minus (119903119886 1003816100381610038161003816119877(1) ) = 119908minus [119875 (119903119886 1003816100381610038161003816119877(1) )]120587+ (119903119886 1003816100381610038161003816119877(119887) ) = 119908+ [ 119898sum

119894=119887

119875 (119903119886 1003816100381610038161003816119877(119894) )]

minus 119908+ [ 119898sum119894=119887+1

119875 (119903119886 1003816100381610038161003816119877(119894) )] 119887 isin 119890 + 1 119890 + 2 sdot sdot sdot 119898 minus 1

120587minus (119903119886 1003816100381610038161003816119877(119887) ) = 119908minus [ 119887sum119894=1

119875 (119903119886 1003816100381610038161003816119877(119894) )]

minus 119908minus [119887minus1sum119894=1

119875 (119903119886 1003816100381610038161003816119877(119894) )] 119887 isin 2 3 sdot sdot sdot 119890

(10)

where 120587+(sdot) and 120587minus(sdot) denote the perceived probability ofdecision-makers corresponding to 119881(119903119886|119877(119887)) ge 0 119887 isin119890 + 1 119890 + 2 sdot sdot sdot 119898 and 119881(119903119886|119877(119887)) le 0 119887 isin 1 2 sdot sdot sdot 119890respectively 119908+(sdot) and 119908minus(sdot) are the following nonlinearfunctions

119908+ (119901) = 119901120574[119901120574 + (1 minus 119901)120574]1120574

119908minus (119901) = 119901120575[119901120575 + (1 minus 119901)120575]1120575

(11)

where 119901 denotes the probability 120574 and 120575 are parametersthat used to describe the curve degree of 119908+(sdot) and 119908minus(sdot)respectively

Step 3 Calculate the overall prospect value of RPP in eachPOC-impacted region

To be convenient for us to compare and discuss the calcu-lated results it is necessary to ensure that different prospectvalues of RPP in different regions have the same upper andlower bounds To do this 119881(119903119886|119877(119887)) is standardized in thefollowing manner

119881 (119903119886 1003816100381610038161003816119877(119887) ) = 119881 (119903119886 1003816100381610038161003816119877(119887) ) minus 119881min (119903119886 1003816100381610038161003816119877(119894) )119881max (119903119886 1003816100381610038161003816119877(119894) ) minus 119881min (119903119886 1003816100381610038161003816119877(119894) ) 119894 isin 1 2 sdot sdot sdot 119898

(12)

where 119881(119903119886|119877(119887)) is the standardized result of 119881(119903119886|119877(119887))obviously 119881(119903119886|119877(119887)) isin [0 1] 119881max(119903119886|119877(119894)) = max119881(119903119886|119877(1))119881(119903119886|119877(2)) sdot sdot sdot 119881(119903119886|119877(119898)) 119881min(119903119886|119877(119894)) = min119881(119903119886|119877(1))119881(119903119886|119877(2)) sdot sdot sdot 119881(119903119886|119877(119898)) Meanwhile the positive correla-tion between 119881(119903119886|119877(119887)) and 119881(119903119886|119877(119887)) can be found

Further the standardized results of 119881(119903119886|119877(119887)) ge 0 119887 isin119890 + 1 119890 + 2 sdot sdot sdot 119898 and 119881(119903119886|119877(119887)) le 0 119887 isin 1 2 sdot sdot sdot 119890are denoted as119881+(119903119886|119877(119887)) and119881minus(119903119886|119877(119887)) respectively Both119881+(119903119886|119877(119887)) and 119881minus(119903119886|119877(119887)) can be obtained by formula (12)obviously 119881+(119903119886|119877(119887)) isin [0 1] 119881minus(119903119886|119877(119887)) isin [0 1]

To sum up it can be seen that119881+(119903119886|119877(119887)) and119881minus(119903119886|119877(119887))correspond to the perceived probabilities 120587+(119903119886|119877(119887)) and120587minus(119903119886|119877(119887)) of decision-makers respectivelyThus the follow-ing formula (13) can be derived by considering the prospectvalue of RPP in all POC-impacted regions

119874119881(119877(119887)) = 119860sum119886=1

[119910120587+ (119903119886 1003816100381610038161003816119877(119887) ) 119881+ (119903119886 1003816100381610038161003816119877(119887) )+ (1 minus 119910) 120587minus (119903119886 1003816100381610038161003816119877(119887) )119881minus (119903119886 1003816100381610038161003816119877(119887) )]

(13)

Where119874119881(119877(119887)) denotes the overall prospect value of RPP inall POC-impacted regions after carrying out the alternativeplan 119877(119887) 119910 is an 0-1 integer variable and satisfies

119910 = 0 119887 isin 1 2 sdot sdot sdot 119890 1 119887 isin 119890 + 1 119890 + 2 sdot sdot sdot 119898 (14)

As mentioned above for determining the optimalresponse plan from alternatives when a certain scenario ofPOC appears decision-makers should focus on the interven-tion effect of the alternatives on RPP Meanwhile combinedwith the proposed method for solving the SRPP in POCwe can see that the intervention effect of an alternative planon RPP is reflected in the overall prospect value of RPP inall POC-impacted regions after carrying out this alternativeTherefore the principle of determining the optimal responseplan is which alternative plan can maximize the value offormula (13) in other words the larger the 119874119881(119877(119887)) is thebetter the alternative plan 119877(119887) is On this basis the ranking ofthe alternatives can be obtained In particular if alternatives1198771198941 and 1198771198942 (1198941 = 1198942) satisfy 119874119881(1198771198941) = 119874119881(1198771198942) thenwhich is better can be determined further by comparingtheir expected cost difficulty in manipulation and start-up timeliness etc Eventually based on the ranking of

10 Complexity

Table 2 The supply of materials during the past period of time the public observe

region 1 region 2 region 3 region 4 region 5119889119886119905 forall119905 isin 1 30 (468) (139) (349) (5611) (234)119889119886119905 forall119905 isin 1 30 6 4 5 7 3119863119860119879 5

Table 3 The importance and ambiguity of POC

1205881119894119895 11986411 11986412 119864131198771 3 5 71198772 2 3 51198773 1 2 3

the alternatives decision-makers can determine the optimalresponse plan in time according to their empirical data (egthe need for related material financial and human resources)learned from the previous emergency drill

4 An Example

Major epidemics of infectious diseases have occurred fre-quently in recent years and they often lead to the POC andthe snap-up of related protective drugs such as SARS in2003 and H1N1 flu in 2009 described above Therefore atheoretical example of the SRPP in POC in a country underthe background of this kind of major epidemics of infectiousdiseases is given to illustrate the potential application andeffectiveness of the proposedmethod Symptomatic incidentsoften occur in the early stages of outbreaks of major epi-demics of infectious diseases such as suspected viral infectorsappear and they have a long-term fever Since this concernsthe safety of the public some panic and rumors have beenpromptly triggered which might as well be recorded asthe current scenario 1198781 faced by decision-makers ie 119878 =1198781 and 119875(1198781) = 1 After estimating the situation of thePOC decision-makers have determined the following threealternatives based on emergency preparedness regulations1198771 Establish a response team tomonitor the developmentof the POC and notify the relevant enterprises to prepare forthe response1198772 Launch guidance strategies for the POC based on theresponse preparation and require enterprises to ensure thesupply of materials the public may snap up1198773 On the basis of the POCguidance inform the progressof dealing with the epidemic event in a timely manner andorder enterprises to increase the supply of relevant materials

After carrying out the alternative plan 119877119894 119894 isin 1 2 3 thecurrent scenario 1198781 will evolve into one of the following newscenarios11986411 Panics of the public are effectively alleviated rumorsare reasonably controlled and the POC is declined or disap-peared11986412 The spreading scope of the POC is only concentratedin a small part of the country and most parts the country arenot impacted

11986413 The POC has an impact on all parts of the countryleading to the spread of serious rumors and panics

In the current scenario 1198781 we set up the total numberof regions the public concern 119860 = 5 and 119886 isin 1 2 3 4 5A day is denoted as the unit of the past period of time 119905the public observe and 119879 = 30 119905 isin 1 2 sdot sdot sdot 30 Thesupply ofmaterials119889119886119905 and its expected value119889119886119905 calculated byformula (9) are shown in Table 2 Meanwhile we can obtain119863119860119879 = 5 by formula (1) The spreading scope 1205881 is set asthe evaluation indicator for the importance and ambiguityof POC and its value ranges from 1 to 7 where 1 and 7denote the minimum and maximum respectively thereby120588 = 1205881 Thus the values of 1205881119894119895 can be obtained as shownin Table 3 Under carrying out different alternative plans thecurrent supply of materials 119889119877119894

119886(119879+1)in the region 119886 is shown

in Table 4 hence we can obtain 1198631198771119860(119879+1)

= 4 1198631198772119860(119879+1)

= 61198631198773119860(119879+1)

= 10 by formula (2) Assuming that the values ofcritical sense 120579119886 of the public range from 1 to 7 where 1 and 7indicate the lowest and highest respectively here we assume120579119886 = 119886 119886 isin 1 2 3 4 5 Considering the completeness ofalternative plans and the severity of new scenarios we canobtain the probability of new scenarios appear by analyzingthe related historical data with neural network Bayesianreasoning evidence reasoning etc here the data given in[26] is used as shown in Table 5 For other parameters it maybe appropriate to set up as 1205721 = 1205722 = 1205731 = 1205732 = 088 1205821 =1205822 = 225 120574 = 061 120575 = 069 [34 35] Based on this thevalue of RPP in each region can be obtained by using formula(6) and setting up 1199030 = 10 as shown in Table 6 In summarythe decision steps for solving the SRPP in the POC with theproposed method are as follows

Step 1 119881(119903119886|1198771198941) and 119881(119903119886|1198771198941) 119886 = 1 2 3 4 5 1198941 = 1 2 3are first obtained with formulae (7) (8) and (12) as shown inTable 7 and Table 8 respectively

Step 2 The prospect values 119881(119903119886|1198771198941) 1198941 = 1 2 3 of RPP indifferent regions are ranked respectively and 119881(119903119886|1198771) lt119881(119903119886|1198772) lt 0 lt 119881(119903119886|1198773) 119886 = 1 2 3 4 5 can be obtainedwhich is recorded as 119881(119903119886|119877(1)) lt 119881(119903119886|119877(2)) lt 0 lt119881(119903119886|119877(3)) 119886 = 1 2 3 4 5 Based on this if we set up theprobability that the RPP takes the current value in the region119886 as 119875119895119886 = 05 119886 = 1 2 3 4 5 119895 = 1 2 3 when 1198781 evolves into1198641119895 after carrying out the alternative plan 119877(119887) then the valueof 119875(119903119886|119877(119887)) can be obtained in conjunction with Table 5that is 119875(119903119886|119877(119887)) = sum3119895=1 119875(1198781)119875(1198641119895 |119877(119887))119875119895119886 = 05 119886 =1 2 3 4 5 119887 = 1 2 3 Further substituting 119875(119903119886|119877(119887)) = 05into formulae (10) and (11) we can obtain 120587minus(119903119886|119877(1)) = 0454120587minus(119903119886|119877(2)) = 0546 120587+(119903119886|119877(3)) = 0421 119886 = 1 2 3 4 5

Complexity 11

Table 4 The supply of materials in the current period of time the public observe

119889119877i119886(119879+1) region 1 region 2 region 3 region 4 region 51198771 4 5 3 6 21198772 5 8 6 7 41198773 7 9 10 13 11

Table 5 The relationship between alternative plans and newscenarios

119875(1198641119895 1003816100381610038161003816119877119894 ) 11986411 11986412 119864131198771 01 03 061198772 02 025 0551198773 03 02 05

Step 3 As shown in Table 8 119881minus(1199031|119877(2)) = 0541 119881minus(1199032|119877(2))= 0568 119881minus(1199033|119877(2)) = 0610 and 119881minus(1199034|119877(2)) = 0643119881minus(119903119886|119877(1)) = 0119881+(119903119886|119877(3)) = 1 119886 = 1 2 3 4 5 Based onthis the overall prospect value of RPP in all POC-impactedregions after carrying out the alternative plan119877(119887) 119887 isin 1 2 3can be obtained by formulae (13) and (14) namely119874119881(119877(1)) =0 119874119881(119877(2)) = 1636 119874119881(119877(3)) = 2105 Therefore accordingto the principle of determining the optimal response plan wecan obtain 119877(3) ≻ 119877(2) ≻ 119877(1) (where ldquo≻rdquo denotes ldquobetterthanrdquo) ie 1198773 ≻ 1198772 ≻ 1198771

In fact the calculation results are also consistent with theactual situation of emergency decision-making for POC inthe context of such a major epidemic of infectious diseasesFor example during the response to the POC caused bySARS in 2003 it is precisely because the characteristics ofSARS virus are not clear and the evolution trend of POCis difficult to be identified effectively in the early stage ofSARS outbreaks that some rumors and panics failed to drawsufficient attention from decision-makers ie which accordswith 1198771 or 1198772 As a result the POC evolves at an amazingspeed and lead to large regions of panic buying of materials(thermometers masks rice vinegar isatis root etc) Untilthe Department of Health of Guangdong Province of Chinaand the CPC Central Committee and State Council one afteranother issued a variety of strong response measures iewhich accords with 1198773 the evolution of the POC is finallysubsided and controlled

5 Conclusions

Themulticase study is first employed to describe the SRPP inPOC that may lead to panic buying of materials derived frompublic emergencies and thus eight typical cases are chosen toanalyze the POC and its relevant response measures Thenthe SRPP is defined in detail that is how to determinethe optimal response plan from the alternatives from theperspective of the intervention of RPP with the consider-ation of behavioral characteristics of decision-makers Todo this the RPP is described with prospect theory throughconsidering the behavioral characteristics and critical sense

of the public the response measures of decision-makers andthe importance and ambiguity of POC Further consideringthe behavioral characteristics of decision-makers and theimpact of alternative plans on the evolution of scenarios ofPOC a new decision method for solving the SRPP with theintervention of the RPP is proposed by using cumulativeprospect theory and a manner of comparing alternativeresponse plans for each other The proposed method showsthat the different overall prospect values of the RPP aftercarrying out different alternative plans can be compared toobtain the ranking of alternatives Finally an example of theSRPP in POC in the context of major epidemics of infectiousdiseases is given to illustrate the potential application andeffectiveness of the proposed method

This study can not only enrich the methodology ofselecting optimal response plan but also provide some the-oretical support for how to restrict the outbreak of secondaryincidents in POC such as the panic buying of materials Tosum up the specific contributions of this study are threefoldFirst the RPP is described with prospect theory throughconsidering the behavioral characteristics and critical senseof the public the response measures of decision-makersand the importance and ambiguity of POC Second a newdecision method for solving the SRPP in POC that may leadto panic buying of materials is proposed from the perspectiveof the intervention of alternatives on RPPThird a manner ofcomparing the alternatives for each other is taken to avoid thedeviation probably caused by setting up the reference pointartificially and the impact of the alternatives on the evolutionof POC scenarios is considered in the proposed method Theabove contributions overcome the shortages that the existingstudies neglect the decision-makersrsquo urgent need for decisionmethods in response to POC that may lead to panic buyingof materials in practice

For the future research there are two directions to beattached great importance to One is that the SRPP with timeconstraint probably needs to be solved further and thus theproposed method in this paper needs to be extended Thereason is that the response time is probably an importantfactor affecting the accuracy of the decision-making whendetermining the optimal plan in response to POC Thisdirection is motivated by the response to natural disastersand environmental emergencies For example Wex et al [13]point out that response plans should be determined within30 minutes after the outbreak of natural disasters In addi-tion according to ldquothe emergency plan for environmentalemergencies in Beijing of Chinardquo (the revised edition in2015) the emergency management office in Beijing stipulatesthat the municipal environmental protection bureau and thedistrict government must submit the general and detailedsituation within 10 minutes and 2 hours respectively after

12 Complexity

Table 6 The relationship between alternative plans and RPP

119903119886 11986411 11986412 119864131198771 28561 16776 12250 9705 9000 42790 24794 18177 14593 12250 56194 32227 23509 18865 159161198772 9621 9000 7629 6989 6613 25557 13215 9000 7959 7371 40025 21683 14742 10664 90001198773 5878 4111 3539 3255 3086 10000 5878 4691 4111 3767 19274 7760 5878 4984 4458

Table 7 The prospect value of RPP

119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 -39953 -26594 -20357 -16735 -140431198772 -8674 -4924 -2835 -1741 -15771198773 17888 11582 8380 6591 5615

Table 8 The standardization for the prospect value of RPP

119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 0 0 0 0 01198772 0541 0568 0610 0643 06341198773 1 1 1 1 1

environmental emergencies happen The other is that theSRPP with dynamic alternative response plans need to bestudied further due to the content and the quantity ofalternatives are probably dynamic or can be updated with theevolution of POC

Data Availability

The data used to support the findings of this study areincluded within this paper It is also available from thecorresponding author upon request

Additional Points

Highlights The RPP is described with prospect theorythrough considering the behavioral characteristics and crit-ical sense of the public the response measures of decision-makers and the importance and ambiguity of POC

A new decision method for solving the SRPP in POC thatmay lead to panic buying of materials is proposed from theperspective of the intervention of alternatives on RPP

A manner of comparing the alternatives for each otheris taken to avoid the deviation probably caused by settingup the reference point artificially and the impact of thealternatives on the evolution of POC scenarios is consideredin the proposed method

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was jointly supported by the National NaturalScience Foundation of China (Project nos 71704001 and

71601002) the Natural Science Foundation in AnhuiProvince (Project nos 1808085QG224 and 1708085MG168)the Humanities and Social Science Key Project ofAnhui Provincial Education Department (Project noSK2019A0075) the Planning Funds of Philosophy and SocialScience in Anhui Province (Project nos AHSKY2018D13 andAHSKQ2016D19) and the Humanities and Social SciencesFoundation of Ministry of Education of China (Project no18YJC630249)

References

[1] S Gupta MK Starr R Z Farahani andNMatinrad ldquoDisastermanagement fromaPOMperspectivemapping a newdomainrdquoProduction andOperationsManagement vol 25 no 10 pp 1611ndash1637 2016

[2] I Akgun F Gumusbuga and B Tansel ldquoRisk based facilitylocation by using fault tree analysis in disaster managementrdquoOmega vol 52 pp 168ndash179 2015

[3] NArgyris and S French ldquoNuclear emergency decision supportA behaviouralORperspectiverdquoEuropean Journal ofOperationalResearch vol 262 no 1 pp 180ndash193 2017

[4] N P Rachaniotis T K Dasaklis and C P Pappis ldquoA deter-ministic resource scheduling model in epidemic control a casestudyrdquo European Journal of Operational Research vol 216 no 1pp 225ndash231 2012

[5] K Kim Y M Baek and N Kim ldquoOnline news diffusiondynamics and public opinion formation A case study of thecontroversy over judgesrsquo personal opinion expression on SNS inKoreardquo Social Science Journal vol 52 no 2 pp 205ndash216 2015

[6] K Fan and W Pedrycz ldquoEmergence and spread of extremistopinionsrdquo Physica A Statistical Mechanics and its Applicationsvol 436 pp 87ndash97 2015

[7] M Xie Z Jia Y Chen and Q Deng ldquoSimulating the spreadingof two competing public opinion information on complexnetworkrdquoAppliedMathematics vol 3 no 9 pp 1074ndash1078 2012

Complexity 13

[8] X Du Y Ye R Y Lau and Y Li ldquoOpinionRings Inferring andvisualizing the opinion tendency of socially connected usersrdquoDecision Support Systems vol 75 pp 11ndash24 2015

[9] NMa andY Liu ldquoSuperedgeRank algorithmand its applicationin identifying opinion leader of online public opinion supernet-workrdquo Expert Systems with Applications vol 41 no 4 pp 1357ndash1368 2014

[10] F Battiston A Cairoli V Nicosia A Baule and V LatoraldquoInterplay between consensus and coherence in a model ofinteracting opinionsrdquo Physica D Nonlinear Phenomena vol323-324 pp 12ndash19 2016

[11] D Centola ldquoThe spread of behavior in an online social networkexperimentrdquo Science vol 329 no 5996 pp 1194ndash1197 2010

[12] K Amailef and J Lu ldquoOntology-supported case-based rea-soning approach for intelligent m-Government emergencyresponse servicesrdquo Decision Support Systems vol 55 no 1 pp79ndash97 2013

[13] F Wex G Schryen S Feuerriegel and D Neumann ldquoEmer-gency response in natural disaster management allocation andscheduling of rescue unitsrdquo European Journal of OperationalResearch vol 235 no 3 pp 697ndash708 2014

[14] R P Hamalainen M R K Lindstedt and K Sinkko ldquoMulti-attribute risk analysis in nuclear emergency managementrdquo RiskAnalysis vol 20 no 4 pp 455ndash467 2000

[15] J Geldermann V Bertsch M Treitz S French K NPapamichail and R P Hamalainen ldquoMulti-criteria decisionsupport and evaluation of strategies for nuclear remediationmanagementrdquoOmega vol 37 no 1 pp 238ndash251 2009

[16] L Yu and K K Lai ldquoA distance-based group decision-makingmethodology for multi-person multi-criteria emergency deci-sion supportrdquo Decision Support Systems vol 51 no 2 pp 307ndash315 2011

[17] F-S Chang J-S Wu C-N Lee and H-C Shen ldquoGreedy-search-based multi-objective genetic algorithm for emergencylogistics schedulingrdquo Expert Systems with Applications vol 41no 6 pp 2947ndash2956 2014

[18] G Fu ldquoA fuzzy optimization method for multicriteria decisionmaking an application to reservoir flood control operationrdquoExpert Systems with Applications vol 34 no 1 pp 145ndash1492008

[19] J Yang and C Xu ldquoEmergency decision engineering modelbased on sequential gamesrdquo Systems Engineering Procedia vol5 pp 276ndash282 2012

[20] Y Liu Z-P Fan Y Yuan and H Li ldquoA FTA-based methodfor risk decision-making in emergency responserdquo Computers ampOperations Research vol 42 pp 49ndash57 2014

[21] A Ebrahim Nejad I Niroomand and O KuzgunkayaldquoResponsive contingency planning in supply risk managementby considering congestion effectsrdquo OMEGA vol 48 pp 19ndash352014

[22] F He and J Zhuang ldquoBalancing pre-disaster preparedness andpost-disaster reliefrdquo European Journal of Operational Researchvol 252 no 1 pp 246ndash256 2016

[23] Y Liu Z-P Fan and Y Zhang ldquoRisk decision analysis inemergency response a method based on cumulative prospecttheoryrdquo Computers amp Operations Research vol 42 pp 75ndash822014

[24] Z Fan Y Liu and R Shen ldquoRisk decision analysis methodfor emergency response based on prospect theoryrdquo SystemsEngineering mdash Theory ampPractice vol 32 no 5 pp 977ndash9842012

[25] S Li N Yang and Y Zhang ldquoRisk decision analysis method foremergency plan selectionrdquoControl and Decision vol 28 no 12pp 1859ndash1863 2013

[26] Z-Y Wang and Y-J Li ldquoInterventionmechanism of behavioraldecision-making on emergency and its starting strategy ofcontingency planrdquo Systems Engineering mdash Theory amp Practicevol 35 no 7 pp 1863ndash1870 2015

[27] ZWang andY Li ldquoStructural description and interaction law ofpublic opinion propagation and emergency decision-makingrdquoSystems Engineering mdash Theory amp Practice vol 35 no 8 pp2064ndash2073 2015

[28] M Frondel M Simora and S Sommer ldquoRisk perception ofclimate change empirical evidence for germanyrdquo EcologicalEconomics vol 137 pp 173ndash183 2017

[29] G Wachinger O Renn C Begg and C Kuhlicke ldquoThe riskperception paradox-implications for governance and commu-nication of natural hazardsrdquo Risk Analysis vol 33 no 6 pp1049ndash1065 2013

[30] L N Rickard ldquoPerception of risk and the attribution ofresponsibility for accidentsrdquoRiskAnalysis vol 34 no 3 pp 514ndash528 2014

[31] T M Lee E M Markowitz P D Howe C-Y Ko and A ALeiserowitz ldquoPredictors of public climate change awareness andrisk perception around the worldrdquo Nature Climate Change vol5 no 11 pp 1014ndash1020 2015

[32] Y Su F Zhao and L Tan ldquoWhether a large disaster couldchange public concern and risk perception a case study ofthe 721 extraordinary rainstorm disaster in Beijing in 2012rdquoNatural Hazards vol 78 no 1 pp 555ndash567 2015

[33] R A Ferrer W M Klein A Persoskie A Avishai-Yitshak andP Sheeran ldquoThe tripartite model of risk perception (TRIRISK)distinguishing deliberative affective and experiential compo-nents of perceived riskrdquo Annals of Behavioral Medicine vol 50no 5 pp 653ndash663 2016

[34] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 pp 263ndash2911979

[35] A Tversky and D Kahneman ldquoAdvances in prospect theorycumulative representation of uncertaintyrdquo Journal of Risk andUncertainty vol 5 no 4 pp 297ndash323 1992

[36] H Bleichrodt U Schmidt and H Zank ldquoAdditive utility inprospect theoryrdquo Management Science vol 55 no 5 pp 863ndash873 2009

[37] M Nagarajan and S Shechter ldquoProspect theory and thenewsvendor problemrdquo Management Science vol 60 no 4 pp1057ndash1062 2014

[38] X Long and J Nasiry ldquoProspect theory explains newsvendorbehavior the role of reference pointsrdquoManagement Science vol61 no 12 pp 3009ndash3012 2015

[39] R M Campos-Vazquez and E Cuilty ldquoThe role of emotionson risk aversion A Prospect Theory experimentrdquo Journal ofBehavioral and Experimental Economics vol 50 pp 1ndash9 2014

[40] X Wang C Ma and J Ruan ldquoEmergency supplies optimalscheduling considering the publicrsquos psychological risk percep-tionrdquo Systems Engineering mdash Theory amp Practice vol 33 no 7pp 1735ndash1742 2013

[41] C Durugbo ldquoCompetitive product-service systems lessonsfrom a multicase studyrdquo International Journal of ProductionResearch vol 51 no 19 pp 5671ndash5682 2013

[42] K M Eisenhardt and M E Graebner ldquoTheory building fromcases opportunities and challengesrdquo Academy of ManagementJournal (AMJ) vol 50 no 1 pp 25ndash32 2007

14 Complexity

[43] Z S Byrne K J Dvorak J M Peters I Ray A Howe andD Sanchez ldquoFrom the userrsquos perspective Perceptions of riskrelative to benefit associatedwith using the InternetrdquoComputersin Human Behavior vol 59 pp 456ndash468 2016

[44] M Setbon J Raude C Fischler and A Flahault ldquoRisk per-ception of the ldquomad cow diseaserdquo in France Determinants andconsequencesrdquo Risk Analysis vol 25 no 4 pp 813ndash826 2005

[45] B Rohrmann ldquoA socio-psychological model for analyzing riskcommunication processesrdquoTheAustralasian Journal of Disasterand Trauma Studies vol 2000 no 2 pp 30ndash50 2000

[46] J Kind W J Wouter Botzen and J C J H Aerts ldquoAccountingfor risk aversion income distribution and social welfare in cost-benefit analysis for flood risk managementrdquo Wiley Interdisci-plinary Reviews Climate Change vol 8 no 2 pp 1ndash20 2017

[47] X Liu Z Wang and S Zhang ldquoA new methodology forhesitant fuzzy emergency decision making with unknownweight informationrdquo Complexity vol 2018 Article ID 514534812 pages 2018

[48] Y Sarafidis ldquoWhat have you done forme lately Release of infor-mation and strategic manipulation of memoriesrdquo EconomicJournal vol 117 no 518 pp 307ndash326 2007

[49] M Abdellaoui H Bleichrodt and C Paraschiv ldquoLoss aversionunder prospect theory a parameter-free measurementrdquo Man-agement Science vol 53 no 10 pp 1659ndash1674 2007

[50] J K Roehrich J Grosvold and S U Hoejmose ldquoReputationalrisks and sustainable supply chain management Decisionmaking under bounded rationalityrdquo International Journal ofOperations and Production Management vol 34 no 5 pp 695ndash719 2014

[51] N J Foss and LWeber ldquoMovingOpportunism to the Back SeatBounded Rationality Costly Conflict and Hierarchical FormsrdquoAcademy ofManagement Review (AMR) vol 41 no 1 pp 61ndash792016

[52] S Chandra and A Aggarwal ldquoOn solving matrix games withpay-offs of triangular fuzzy numbers Certain observations andgeneralizationsrdquo European Journal of Operational Research vol246 no 2 pp 575ndash581 2015

[53] G W Allport and L PostmanThe Psychology of Rumor HenryHolt Oxford UK 1947

[54] M Saaksjarvi T Gill and E J Hultink ldquoHow rumors andpreannouncements foster curiosity toward productsrdquo EuropeanJournal of Innovation Management vol 20 no 3 pp 350ndash3712017

[55] A Chorus ldquoThe basic law of rumorrdquoThe Journal of Abnormaland Social Psychology vol 48 no 2 pp 313-314 1953

[56] S Elizabeth and L Sujatha ldquoProject scheduling method usingtriangular intuitionistic fuzzy numbers and triangular fuzzynumbersrdquo Applied Mathematical Sciences vol 9 no 1ndash4 pp185ndash198 2015

[57] T-S Liou and M J Wang ldquoRanking fuzzy numbers withintegral valuerdquo Computers in Human Behavior vol 50 no 3pp 247ndash255 1992

[58] R R Yager ldquoA procedure for ordering fuzzy subsets of the unitintervalrdquo Information Sciences vol 24 no 2 pp 143ndash161 1981

Hindawiwwwhindawicom Volume 2018

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Applied MathematicsJournal of


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International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Dierential EquationsInternational Journal of

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Decision SciencesAdvances in


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Submit your manuscripts atwwwhindawicom

2 Complexity

A large number of cases show that POC usually occursbecause of public opinion on public emergencies themselvesand decision-makersrsquo response measures (such as the above-mentioned Chinarsquos SARS in 2003 and Japanrsquos nuclear leakagein 2011) after outbreaks of public emergencies Thereforesimilar to public emergencies POC has gradually becomean important research direction in academia focusing onevolution process modeling [6 7] evolution path exploration[8] opinion leader identification [9] and influencing factorsanalysis [10 11] Accordingly as an important means fordecision-makers to deal with social risks or public crisesthe selection or evaluation methods of alternative responseplans have attracted much attention of scholars [12ndash22]Amailef and Lu [12] proposed an ontology-based case-based reasoning (OS-CBR)method which provides a feasiblestrategy for the evaluation and start-up of decision supportsystems in emergency situations by constructing a caseretrieval process Wex et al [13] analyzed the effect of thedistribution efficiency of rescue materials on reducing theharm of natural disasters Then by weighting the materialsaccording to their importance a decision support model wasdesigned to minimize the total distribution time and thestarting strategy of the material distribution plan was givenHamalainen et al [14] proposed a risk analysis method basedon multi-attribute utility theory for screening the optimalresponse plan in emergency activities of nuclear accidentsGeldermann et al [15] proposed a multicriteria decisionsupport method for solving the optimal selection of multiplefeasible alternatives in nuclear or radiological accidentsFor decision-making of multicriteria and multiperson inunconventional emergencies Yu and Lai [16] proposed adistance-based group decision making method (GDM) Toquickly and effectively relieve the victims of disasters Changet al [17] proposed a greedy search andmultiobjective geneticalgorithm (GSMOGA) to adjust the allocation of availableresources and generate a feasible emergency logistics schedul-ing scheme For solving the multicriteria decision-makingproblem in fuzzy environment Fu [18] proposed a fuzzyoptimization method and applied it to the scheme selectionof reservoir flood control Yang and Xu [19] establishedan engineering model of the interaction between decision-makers and emergencies by means of sequential gameswhich was used to generate the optimal relief scheme foremergencies Liu et al [20] proposed a fault tree analysis-(FTA-) based evaluation method of response schemes andapplied it to the response of H1N1 infectious diseases Aimingat the interruption risk of supply chain caused by naturaldisasters Nejad et al [21] suggested a mixed integer pro-gramming model to study the emergency preparedness ofsuppliers in the process of improving emergency responsecapability of the supply chain He and Zhuang [22] pointedout that the hazard is determined by both preparation andrescue and thus they studied the allocation of emergencypreparedness and rescue costs by constructing a two-stagedynamic programming model In addition some scholarsbelieve that considering the behavioral characteristics ofdecision-makers in the selection of alternatives is helpful toimprove the reliability of decision-making [23ndash26] Liu et al[23] Fan et al [24] and Li et al [25] considered the behavioral

characteristics (loss aversion reference dependence dimin-ishing sensitivity etc) of decision-makers and the impactof the alternatives on hazards of emergency scenarios andapplied prospect theory to study the selection of alternativesWang and Li [26] put forward a set description method ofbehavioral decision-making in emergencies and then studiedthe selection of alternatives with cumulative prospect theoryby considering the evaluation of scenario attributes and theintervention of alternatives on both scenario hazards andscenario evolution

To sum up the existing studies have made significantcontributions to the identification of the evolution law ofpublic emergencies and POC and the selection of alternativeresponse plans However it is worth mentioning that most ofthe abovementioned achievements on the decision methodfor selecting the optimal response plan from alternatives arebased on public emergencies little attention has been paidto decision-makersrsquo urgent need for decision methods inresponse to POC that may lead to panic buying of materials(eg during the POCon ldquoseawater pollutionrdquo and ldquoiodine canresist nuclear radiationrdquo triggered by Japanrsquos nuclear leakagein 2011 the Chinese Government adopted a response planthat combines measures such as holding press conferencespunishing lawbreakers and increasing the supply of relatedmaterials) Different frompublic emergencies POC generallyhas clear promoters and media channels and its evolutionprocess (including appearance development climax declineand disappearance [27]) is essentially the change process ofbehavioral relationships between the public (disseminatorrecipient etc) From the perspective of cognitive psychologythe deep-seated cause for the formation of this change processis the change of RPP [28] In fact the change of RPPis synchronized with the change of POC to some extentbecause the level of RPP directly determines the intensityof POC and the intensity of POC directly reflects the levelof RPP So far the description of RPP has been widelyconcerned by scholars and the related studies are mainlybased on sociocultural paradigmand psychometric paradigm[29ndash33] but there is no quantitative model of RPP in thecontext of POC Prospect theory and its improved version(eg cumulative prospect theory) [34 35] have attractedmuch attention in recent years because of their ability todescribe some human decision-making behaviors in risk oruncertainty situations (loss aversion reference dependencydiminishing sensitivity etc) [36ndash38] In the field of emer-gency management prospect theory has been applied tothe public level [39 40] in addition to the decision-makerslevel [23ndash26] described above Campos-Vazquez and Cuilty[39] used prospect theory to measure the impact of studentemotions on the risk and loss aversion parameters whenmanipulated by drug violence and youth unemploymentin Mexico City Wang et al [40] used prospect theory todescribe the RPP on the arrival time of emergency materialsin the study of optimal scheduling problem of emergencymaterials under emergencies Different from these previousstudies this paper will use prospect theory to describe theperceived value of the public on the supply of materialsand then deduce the change and formula of RPP afteranalyzing the behavioral characteristics and critical sense of

Complexity 3










4 Complexity





pneumoniardquo





materials



rumorsrdquo

































information



shortagesrdquo


and robbery of food














flashlights



Complexity 5

POC



emergencies











Response





119863119860119879 = 1119860119879119860sum119886=1

119879sum119905=1

119889119886119905 (1)


119863119860(119879+1) = 1119860119860sum119886=1

119889119886(119879+1) (2)


V (119863119860(119879+1))


(3)


6 Complexity

(DA(T+1) )

DA(T+1)

DAT


DA(T+1)

DAT

r(DA(T+1) )

r0





= (119863119860(119879+1) minus 120588120579119863119860119879)


1205731 119863119860(119879+1) minus 120588120579119863119860119879 lt 0(4)




Complexity 7



119903 (119863119860(119879+1) 120588 120579)

= 1199030 minus (119863119860(119879+1) minus 120588120579119863119860119879)


1205731 119863119860(119879+1) minus 120588120579119863119860119879 lt 0(5)





119860(119879+1) 1198631198772119860(119879+1)

sdot sdot sdot 119863119877119898119860(119879+1)






value is 120588119896119894119895 = sum119892ℎ=1

120588119896119894119895ℎ




8 Complexity

119903119886 (119863119877119894119860(119879+1) 120588119896119894119895 120579119886) =

1199030 minus (119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879)

1205721 119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879 ge 0

1199030 + 1205821 [minus(119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879)]

1205731 119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879 lt 0

(6)



119881(119903119886 100381610038161003816100381610038161198771198941 ) = 1119899 (119898 minus 1)sdot 119899sum119896=1

119898sum1198942=11198942 =1198941

119897sum1198951=1

119897sum1198952=1

119889 [119903119886 (1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) 119903119886 (1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886)]119875 (119878119896) 119875 (1198641198961198951 100381610038161003816100381610038161198771198941 ) 119875 (1198641198961198952 100381610038161003816100381610038161198771198942 )(7)



119889 [119903119886 (1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) 119903119886 (1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886)]


(8)

where 119903119886(1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) minus 119903119886(119863R1198942119860(119879+1)



119889119886119905 = (1 minus 120583) 1198891119886119905 + 1198892119886119905 + 12058311988931198861199052 (9)



Complexity 9



120587+ (119903119886 1003816100381610038161003816119877(119898) ) = 119908+ [119875 (119903119886 1003816100381610038161003816119877(119898) )]120587minus (119903119886 1003816100381610038161003816119877(1) ) = 119908minus [119875 (119903119886 1003816100381610038161003816119877(1) )]120587+ (119903119886 1003816100381610038161003816119877(119887) ) = 119908+ [ 119898sum

119894=119887

119875 (119903119886 1003816100381610038161003816119877(119894) )]

minus 119908+ [ 119898sum119894=119887+1


120587minus (119903119886 1003816100381610038161003816119877(119887) ) = 119908minus [ 119887sum119894=1

119875 (119903119886 1003816100381610038161003816119877(119894) )]



(10)


119908+ (119901) = 119901120574[119901120574 + (1 minus 119901)120574]1120574

119908minus (119901) = 119901120575[119901120575 + (1 minus 119901)120575]1120575

(11)





(12)




119874119881(119877(119887)) = 119860sum119886=1


(13)




10 Complexity




1205881119894119895 11986411 11986412 119864131198771 3 5 71198772 2 3 51198773 1 2 3


4 An Example







= 4 1198631198772119860(119879+1)

= 61198631198773119860(119879+1)




Complexity 11




119875(1198641119895 1003816100381610038161003816119877119894 ) 11986411 11986412 119864131198771 01 03 061198772 02 025 0551198773 03 02 05



5 Conclusions





12 Complexity


119903119886 11986411 11986412 119864131198771 28561 16776 12250 9705 9000 42790 24794 18177 14593 12250 56194 32227 23509 18865 159161198772 9621 9000 7629 6989 6613 25557 13215 9000 7959 7371 40025 21683 14742 10664 90001198773 5878 4111 3539 3255 3086 10000 5878 4691 4111 3767 19274 7760 5878 4984 4458


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 -39953 -26594 -20357 -16735 -140431198772 -8674 -4924 -2835 -1741 -15771198773 17888 11582 8380 6591 5615


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 0 0 0 0 01198772 0541 0568 0610 0643 06341198773 1 1 1 1 1


Data Availability


Additional Points






Acknowledgments



References








Complexity 13




































14 Complexity
























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 3










4 Complexity





pneumoniardquo





materials



rumorsrdquo

































information



shortagesrdquo


and robbery of food














flashlights



Complexity 5

POC



emergencies











Response





119863119860119879 = 1119860119879119860sum119886=1

119879sum119905=1

119889119886119905 (1)


119863119860(119879+1) = 1119860119860sum119886=1

119889119886(119879+1) (2)


V (119863119860(119879+1))


(3)


6 Complexity

(DA(T+1) )

DA(T+1)

DAT


DA(T+1)

DAT

r(DA(T+1) )

r0





= (119863119860(119879+1) minus 120588120579119863119860119879)


1205731 119863119860(119879+1) minus 120588120579119863119860119879 lt 0(4)




Complexity 7



119903 (119863119860(119879+1) 120588 120579)

= 1199030 minus (119863119860(119879+1) minus 120588120579119863119860119879)


1205731 119863119860(119879+1) minus 120588120579119863119860119879 lt 0(5)





119860(119879+1) 1198631198772119860(119879+1)

sdot sdot sdot 119863119877119898119860(119879+1)






value is 120588119896119894119895 = sum119892ℎ=1

120588119896119894119895ℎ




8 Complexity

119903119886 (119863119877119894119860(119879+1) 120588119896119894119895 120579119886) =

1199030 minus (119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879)

1205721 119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879 ge 0

1199030 + 1205821 [minus(119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879)]

1205731 119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879 lt 0

(6)



119881(119903119886 100381610038161003816100381610038161198771198941 ) = 1119899 (119898 minus 1)sdot 119899sum119896=1

119898sum1198942=11198942 =1198941

119897sum1198951=1

119897sum1198952=1

119889 [119903119886 (1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) 119903119886 (1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886)]119875 (119878119896) 119875 (1198641198961198951 100381610038161003816100381610038161198771198941 ) 119875 (1198641198961198952 100381610038161003816100381610038161198771198942 )(7)



119889 [119903119886 (1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) 119903119886 (1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886)]


(8)

where 119903119886(1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) minus 119903119886(119863R1198942119860(119879+1)



119889119886119905 = (1 minus 120583) 1198891119886119905 + 1198892119886119905 + 12058311988931198861199052 (9)



Complexity 9



120587+ (119903119886 1003816100381610038161003816119877(119898) ) = 119908+ [119875 (119903119886 1003816100381610038161003816119877(119898) )]120587minus (119903119886 1003816100381610038161003816119877(1) ) = 119908minus [119875 (119903119886 1003816100381610038161003816119877(1) )]120587+ (119903119886 1003816100381610038161003816119877(119887) ) = 119908+ [ 119898sum

119894=119887

119875 (119903119886 1003816100381610038161003816119877(119894) )]

minus 119908+ [ 119898sum119894=119887+1


120587minus (119903119886 1003816100381610038161003816119877(119887) ) = 119908minus [ 119887sum119894=1

119875 (119903119886 1003816100381610038161003816119877(119894) )]



(10)


119908+ (119901) = 119901120574[119901120574 + (1 minus 119901)120574]1120574

119908minus (119901) = 119901120575[119901120575 + (1 minus 119901)120575]1120575

(11)





(12)




119874119881(119877(119887)) = 119860sum119886=1


(13)




10 Complexity




1205881119894119895 11986411 11986412 119864131198771 3 5 71198772 2 3 51198773 1 2 3


4 An Example







= 4 1198631198772119860(119879+1)

= 61198631198773119860(119879+1)




Complexity 11




119875(1198641119895 1003816100381610038161003816119877119894 ) 11986411 11986412 119864131198771 01 03 061198772 02 025 0551198773 03 02 05



5 Conclusions





12 Complexity


119903119886 11986411 11986412 119864131198771 28561 16776 12250 9705 9000 42790 24794 18177 14593 12250 56194 32227 23509 18865 159161198772 9621 9000 7629 6989 6613 25557 13215 9000 7959 7371 40025 21683 14742 10664 90001198773 5878 4111 3539 3255 3086 10000 5878 4691 4111 3767 19274 7760 5878 4984 4458


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 -39953 -26594 -20357 -16735 -140431198772 -8674 -4924 -2835 -1741 -15771198773 17888 11582 8380 6591 5615


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 0 0 0 0 01198772 0541 0568 0610 0643 06341198773 1 1 1 1 1


Data Availability


Additional Points






Acknowledgments



References








Complexity 13




































14 Complexity
























Journal of












Journal of







Volume 2018






Volume 2018








4 Complexity





pneumoniardquo





materials



rumorsrdquo

































information



shortagesrdquo


and robbery of food














flashlights



Complexity 5

POC



emergencies











Response





119863119860119879 = 1119860119879119860sum119886=1

119879sum119905=1

119889119886119905 (1)


119863119860(119879+1) = 1119860119860sum119886=1

119889119886(119879+1) (2)


V (119863119860(119879+1))


(3)


6 Complexity

(DA(T+1) )

DA(T+1)

DAT


DA(T+1)

DAT

r(DA(T+1) )

r0





= (119863119860(119879+1) minus 120588120579119863119860119879)


1205731 119863119860(119879+1) minus 120588120579119863119860119879 lt 0(4)




Complexity 7



119903 (119863119860(119879+1) 120588 120579)

= 1199030 minus (119863119860(119879+1) minus 120588120579119863119860119879)


1205731 119863119860(119879+1) minus 120588120579119863119860119879 lt 0(5)





119860(119879+1) 1198631198772119860(119879+1)

sdot sdot sdot 119863119877119898119860(119879+1)






value is 120588119896119894119895 = sum119892ℎ=1

120588119896119894119895ℎ




8 Complexity

119903119886 (119863119877119894119860(119879+1) 120588119896119894119895 120579119886) =

1199030 minus (119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879)

1205721 119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879 ge 0

1199030 + 1205821 [minus(119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879)]

1205731 119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879 lt 0

(6)



119881(119903119886 100381610038161003816100381610038161198771198941 ) = 1119899 (119898 minus 1)sdot 119899sum119896=1

119898sum1198942=11198942 =1198941

119897sum1198951=1

119897sum1198952=1

119889 [119903119886 (1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) 119903119886 (1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886)]119875 (119878119896) 119875 (1198641198961198951 100381610038161003816100381610038161198771198941 ) 119875 (1198641198961198952 100381610038161003816100381610038161198771198942 )(7)



119889 [119903119886 (1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) 119903119886 (1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886)]


(8)

where 119903119886(1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) minus 119903119886(119863R1198942119860(119879+1)



119889119886119905 = (1 minus 120583) 1198891119886119905 + 1198892119886119905 + 12058311988931198861199052 (9)



Complexity 9



120587+ (119903119886 1003816100381610038161003816119877(119898) ) = 119908+ [119875 (119903119886 1003816100381610038161003816119877(119898) )]120587minus (119903119886 1003816100381610038161003816119877(1) ) = 119908minus [119875 (119903119886 1003816100381610038161003816119877(1) )]120587+ (119903119886 1003816100381610038161003816119877(119887) ) = 119908+ [ 119898sum

119894=119887

119875 (119903119886 1003816100381610038161003816119877(119894) )]

minus 119908+ [ 119898sum119894=119887+1


120587minus (119903119886 1003816100381610038161003816119877(119887) ) = 119908minus [ 119887sum119894=1

119875 (119903119886 1003816100381610038161003816119877(119894) )]



(10)


119908+ (119901) = 119901120574[119901120574 + (1 minus 119901)120574]1120574

119908minus (119901) = 119901120575[119901120575 + (1 minus 119901)120575]1120575

(11)





(12)




119874119881(119877(119887)) = 119860sum119886=1


(13)




10 Complexity




1205881119894119895 11986411 11986412 119864131198771 3 5 71198772 2 3 51198773 1 2 3


4 An Example







= 4 1198631198772119860(119879+1)

= 61198631198773119860(119879+1)




Complexity 11




119875(1198641119895 1003816100381610038161003816119877119894 ) 11986411 11986412 119864131198771 01 03 061198772 02 025 0551198773 03 02 05



5 Conclusions





12 Complexity


119903119886 11986411 11986412 119864131198771 28561 16776 12250 9705 9000 42790 24794 18177 14593 12250 56194 32227 23509 18865 159161198772 9621 9000 7629 6989 6613 25557 13215 9000 7959 7371 40025 21683 14742 10664 90001198773 5878 4111 3539 3255 3086 10000 5878 4691 4111 3767 19274 7760 5878 4984 4458


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 -39953 -26594 -20357 -16735 -140431198772 -8674 -4924 -2835 -1741 -15771198773 17888 11582 8380 6591 5615


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 0 0 0 0 01198772 0541 0568 0610 0643 06341198773 1 1 1 1 1


Data Availability


Additional Points






Acknowledgments



References








Complexity 13




































14 Complexity
























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 5

POC



emergencies











Response





119863119860119879 = 1119860119879119860sum119886=1

119879sum119905=1

119889119886119905 (1)


119863119860(119879+1) = 1119860119860sum119886=1

119889119886(119879+1) (2)


V (119863119860(119879+1))


(3)


6 Complexity

(DA(T+1) )

DA(T+1)

DAT


DA(T+1)

DAT

r(DA(T+1) )

r0





= (119863119860(119879+1) minus 120588120579119863119860119879)


1205731 119863119860(119879+1) minus 120588120579119863119860119879 lt 0(4)




Complexity 7



119903 (119863119860(119879+1) 120588 120579)

= 1199030 minus (119863119860(119879+1) minus 120588120579119863119860119879)


1205731 119863119860(119879+1) minus 120588120579119863119860119879 lt 0(5)





119860(119879+1) 1198631198772119860(119879+1)

sdot sdot sdot 119863119877119898119860(119879+1)






value is 120588119896119894119895 = sum119892ℎ=1

120588119896119894119895ℎ




8 Complexity

119903119886 (119863119877119894119860(119879+1) 120588119896119894119895 120579119886) =

1199030 minus (119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879)

1205721 119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879 ge 0

1199030 + 1205821 [minus(119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879)]

1205731 119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879 lt 0

(6)



119881(119903119886 100381610038161003816100381610038161198771198941 ) = 1119899 (119898 minus 1)sdot 119899sum119896=1

119898sum1198942=11198942 =1198941

119897sum1198951=1

119897sum1198952=1

119889 [119903119886 (1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) 119903119886 (1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886)]119875 (119878119896) 119875 (1198641198961198951 100381610038161003816100381610038161198771198941 ) 119875 (1198641198961198952 100381610038161003816100381610038161198771198942 )(7)



119889 [119903119886 (1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) 119903119886 (1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886)]


(8)

where 119903119886(1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) minus 119903119886(119863R1198942119860(119879+1)



119889119886119905 = (1 minus 120583) 1198891119886119905 + 1198892119886119905 + 12058311988931198861199052 (9)



Complexity 9



120587+ (119903119886 1003816100381610038161003816119877(119898) ) = 119908+ [119875 (119903119886 1003816100381610038161003816119877(119898) )]120587minus (119903119886 1003816100381610038161003816119877(1) ) = 119908minus [119875 (119903119886 1003816100381610038161003816119877(1) )]120587+ (119903119886 1003816100381610038161003816119877(119887) ) = 119908+ [ 119898sum

119894=119887

119875 (119903119886 1003816100381610038161003816119877(119894) )]

minus 119908+ [ 119898sum119894=119887+1


120587minus (119903119886 1003816100381610038161003816119877(119887) ) = 119908minus [ 119887sum119894=1

119875 (119903119886 1003816100381610038161003816119877(119894) )]



(10)


119908+ (119901) = 119901120574[119901120574 + (1 minus 119901)120574]1120574

119908minus (119901) = 119901120575[119901120575 + (1 minus 119901)120575]1120575

(11)





(12)




119874119881(119877(119887)) = 119860sum119886=1


(13)




10 Complexity




1205881119894119895 11986411 11986412 119864131198771 3 5 71198772 2 3 51198773 1 2 3


4 An Example







= 4 1198631198772119860(119879+1)

= 61198631198773119860(119879+1)




Complexity 11




119875(1198641119895 1003816100381610038161003816119877119894 ) 11986411 11986412 119864131198771 01 03 061198772 02 025 0551198773 03 02 05



5 Conclusions





12 Complexity


119903119886 11986411 11986412 119864131198771 28561 16776 12250 9705 9000 42790 24794 18177 14593 12250 56194 32227 23509 18865 159161198772 9621 9000 7629 6989 6613 25557 13215 9000 7959 7371 40025 21683 14742 10664 90001198773 5878 4111 3539 3255 3086 10000 5878 4691 4111 3767 19274 7760 5878 4984 4458


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 -39953 -26594 -20357 -16735 -140431198772 -8674 -4924 -2835 -1741 -15771198773 17888 11582 8380 6591 5615


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 0 0 0 0 01198772 0541 0568 0610 0643 06341198773 1 1 1 1 1


Data Availability


Additional Points






Acknowledgments



References








Complexity 13




































14 Complexity
























Journal of












Journal of







Volume 2018






Volume 2018








6 Complexity

(DA(T+1) )

DA(T+1)

DAT


DA(T+1)

DAT

r(DA(T+1) )

r0





= (119863119860(119879+1) minus 120588120579119863119860119879)


1205731 119863119860(119879+1) minus 120588120579119863119860119879 lt 0(4)




Complexity 7



119903 (119863119860(119879+1) 120588 120579)

= 1199030 minus (119863119860(119879+1) minus 120588120579119863119860119879)


1205731 119863119860(119879+1) minus 120588120579119863119860119879 lt 0(5)





119860(119879+1) 1198631198772119860(119879+1)

sdot sdot sdot 119863119877119898119860(119879+1)






value is 120588119896119894119895 = sum119892ℎ=1

120588119896119894119895ℎ




8 Complexity

119903119886 (119863119877119894119860(119879+1) 120588119896119894119895 120579119886) =

1199030 minus (119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879)

1205721 119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879 ge 0

1199030 + 1205821 [minus(119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879)]

1205731 119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879 lt 0

(6)



119881(119903119886 100381610038161003816100381610038161198771198941 ) = 1119899 (119898 minus 1)sdot 119899sum119896=1

119898sum1198942=11198942 =1198941

119897sum1198951=1

119897sum1198952=1

119889 [119903119886 (1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) 119903119886 (1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886)]119875 (119878119896) 119875 (1198641198961198951 100381610038161003816100381610038161198771198941 ) 119875 (1198641198961198952 100381610038161003816100381610038161198771198942 )(7)



119889 [119903119886 (1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) 119903119886 (1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886)]


(8)

where 119903119886(1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) minus 119903119886(119863R1198942119860(119879+1)



119889119886119905 = (1 minus 120583) 1198891119886119905 + 1198892119886119905 + 12058311988931198861199052 (9)



Complexity 9



120587+ (119903119886 1003816100381610038161003816119877(119898) ) = 119908+ [119875 (119903119886 1003816100381610038161003816119877(119898) )]120587minus (119903119886 1003816100381610038161003816119877(1) ) = 119908minus [119875 (119903119886 1003816100381610038161003816119877(1) )]120587+ (119903119886 1003816100381610038161003816119877(119887) ) = 119908+ [ 119898sum

119894=119887

119875 (119903119886 1003816100381610038161003816119877(119894) )]

minus 119908+ [ 119898sum119894=119887+1


120587minus (119903119886 1003816100381610038161003816119877(119887) ) = 119908minus [ 119887sum119894=1

119875 (119903119886 1003816100381610038161003816119877(119894) )]



(10)


119908+ (119901) = 119901120574[119901120574 + (1 minus 119901)120574]1120574

119908minus (119901) = 119901120575[119901120575 + (1 minus 119901)120575]1120575

(11)





(12)




119874119881(119877(119887)) = 119860sum119886=1


(13)




10 Complexity




1205881119894119895 11986411 11986412 119864131198771 3 5 71198772 2 3 51198773 1 2 3


4 An Example







= 4 1198631198772119860(119879+1)

= 61198631198773119860(119879+1)




Complexity 11




119875(1198641119895 1003816100381610038161003816119877119894 ) 11986411 11986412 119864131198771 01 03 061198772 02 025 0551198773 03 02 05



5 Conclusions





12 Complexity


119903119886 11986411 11986412 119864131198771 28561 16776 12250 9705 9000 42790 24794 18177 14593 12250 56194 32227 23509 18865 159161198772 9621 9000 7629 6989 6613 25557 13215 9000 7959 7371 40025 21683 14742 10664 90001198773 5878 4111 3539 3255 3086 10000 5878 4691 4111 3767 19274 7760 5878 4984 4458


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 -39953 -26594 -20357 -16735 -140431198772 -8674 -4924 -2835 -1741 -15771198773 17888 11582 8380 6591 5615


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 0 0 0 0 01198772 0541 0568 0610 0643 06341198773 1 1 1 1 1


Data Availability


Additional Points






Acknowledgments



References








Complexity 13




































14 Complexity
























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 7



119903 (119863119860(119879+1) 120588 120579)

= 1199030 minus (119863119860(119879+1) minus 120588120579119863119860119879)


1205731 119863119860(119879+1) minus 120588120579119863119860119879 lt 0(5)





119860(119879+1) 1198631198772119860(119879+1)

sdot sdot sdot 119863119877119898119860(119879+1)






value is 120588119896119894119895 = sum119892ℎ=1

120588119896119894119895ℎ




8 Complexity

119903119886 (119863119877119894119860(119879+1) 120588119896119894119895 120579119886) =

1199030 minus (119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879)

1205721 119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879 ge 0

1199030 + 1205821 [minus(119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879)]

1205731 119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879 lt 0

(6)



119881(119903119886 100381610038161003816100381610038161198771198941 ) = 1119899 (119898 minus 1)sdot 119899sum119896=1

119898sum1198942=11198942 =1198941

119897sum1198951=1

119897sum1198952=1

119889 [119903119886 (1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) 119903119886 (1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886)]119875 (119878119896) 119875 (1198641198961198951 100381610038161003816100381610038161198771198941 ) 119875 (1198641198961198952 100381610038161003816100381610038161198771198942 )(7)



119889 [119903119886 (1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) 119903119886 (1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886)]


(8)

where 119903119886(1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) minus 119903119886(119863R1198942119860(119879+1)



119889119886119905 = (1 minus 120583) 1198891119886119905 + 1198892119886119905 + 12058311988931198861199052 (9)



Complexity 9



120587+ (119903119886 1003816100381610038161003816119877(119898) ) = 119908+ [119875 (119903119886 1003816100381610038161003816119877(119898) )]120587minus (119903119886 1003816100381610038161003816119877(1) ) = 119908minus [119875 (119903119886 1003816100381610038161003816119877(1) )]120587+ (119903119886 1003816100381610038161003816119877(119887) ) = 119908+ [ 119898sum

119894=119887

119875 (119903119886 1003816100381610038161003816119877(119894) )]

minus 119908+ [ 119898sum119894=119887+1


120587minus (119903119886 1003816100381610038161003816119877(119887) ) = 119908minus [ 119887sum119894=1

119875 (119903119886 1003816100381610038161003816119877(119894) )]



(10)


119908+ (119901) = 119901120574[119901120574 + (1 minus 119901)120574]1120574

119908minus (119901) = 119901120575[119901120575 + (1 minus 119901)120575]1120575

(11)





(12)




119874119881(119877(119887)) = 119860sum119886=1


(13)




10 Complexity




1205881119894119895 11986411 11986412 119864131198771 3 5 71198772 2 3 51198773 1 2 3


4 An Example







= 4 1198631198772119860(119879+1)

= 61198631198773119860(119879+1)




Complexity 11




119875(1198641119895 1003816100381610038161003816119877119894 ) 11986411 11986412 119864131198771 01 03 061198772 02 025 0551198773 03 02 05



5 Conclusions





12 Complexity


119903119886 11986411 11986412 119864131198771 28561 16776 12250 9705 9000 42790 24794 18177 14593 12250 56194 32227 23509 18865 159161198772 9621 9000 7629 6989 6613 25557 13215 9000 7959 7371 40025 21683 14742 10664 90001198773 5878 4111 3539 3255 3086 10000 5878 4691 4111 3767 19274 7760 5878 4984 4458


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 -39953 -26594 -20357 -16735 -140431198772 -8674 -4924 -2835 -1741 -15771198773 17888 11582 8380 6591 5615


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 0 0 0 0 01198772 0541 0568 0610 0643 06341198773 1 1 1 1 1


Data Availability


Additional Points






Acknowledgments



References








Complexity 13




































14 Complexity
























Journal of












Journal of







Volume 2018






Volume 2018








8 Complexity

119903119886 (119863119877119894119860(119879+1) 120588119896119894119895 120579119886) =

1199030 minus (119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879)

1205721 119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879 ge 0

1199030 + 1205821 [minus(119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879)]

1205731 119863119877119894119860(119879+1)

minus 120588119896119894119895120579119886 119863119860119879 lt 0

(6)



119881(119903119886 100381610038161003816100381610038161198771198941 ) = 1119899 (119898 minus 1)sdot 119899sum119896=1

119898sum1198942=11198942 =1198941

119897sum1198951=1

119897sum1198952=1

119889 [119903119886 (1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) 119903119886 (1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886)]119875 (119878119896) 119875 (1198641198961198951 100381610038161003816100381610038161198771198941 ) 119875 (1198641198961198952 100381610038161003816100381610038161198771198942 )(7)



119889 [119903119886 (1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) 119903119886 (1198631198771198942119860(119879+1) 12058811989611989421198952 120579119886)]


(8)

where 119903119886(1198631198771198941119860(119879+1) 12058811989611989411198951 120579119886) minus 119903119886(119863R1198942119860(119879+1)



119889119886119905 = (1 minus 120583) 1198891119886119905 + 1198892119886119905 + 12058311988931198861199052 (9)



Complexity 9



120587+ (119903119886 1003816100381610038161003816119877(119898) ) = 119908+ [119875 (119903119886 1003816100381610038161003816119877(119898) )]120587minus (119903119886 1003816100381610038161003816119877(1) ) = 119908minus [119875 (119903119886 1003816100381610038161003816119877(1) )]120587+ (119903119886 1003816100381610038161003816119877(119887) ) = 119908+ [ 119898sum

119894=119887

119875 (119903119886 1003816100381610038161003816119877(119894) )]

minus 119908+ [ 119898sum119894=119887+1


120587minus (119903119886 1003816100381610038161003816119877(119887) ) = 119908minus [ 119887sum119894=1

119875 (119903119886 1003816100381610038161003816119877(119894) )]



(10)


119908+ (119901) = 119901120574[119901120574 + (1 minus 119901)120574]1120574

119908minus (119901) = 119901120575[119901120575 + (1 minus 119901)120575]1120575

(11)





(12)




119874119881(119877(119887)) = 119860sum119886=1


(13)




10 Complexity




1205881119894119895 11986411 11986412 119864131198771 3 5 71198772 2 3 51198773 1 2 3


4 An Example







= 4 1198631198772119860(119879+1)

= 61198631198773119860(119879+1)




Complexity 11




119875(1198641119895 1003816100381610038161003816119877119894 ) 11986411 11986412 119864131198771 01 03 061198772 02 025 0551198773 03 02 05



5 Conclusions





12 Complexity


119903119886 11986411 11986412 119864131198771 28561 16776 12250 9705 9000 42790 24794 18177 14593 12250 56194 32227 23509 18865 159161198772 9621 9000 7629 6989 6613 25557 13215 9000 7959 7371 40025 21683 14742 10664 90001198773 5878 4111 3539 3255 3086 10000 5878 4691 4111 3767 19274 7760 5878 4984 4458


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 -39953 -26594 -20357 -16735 -140431198772 -8674 -4924 -2835 -1741 -15771198773 17888 11582 8380 6591 5615


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 0 0 0 0 01198772 0541 0568 0610 0643 06341198773 1 1 1 1 1


Data Availability


Additional Points






Acknowledgments



References








Complexity 13




































14 Complexity
























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 9



120587+ (119903119886 1003816100381610038161003816119877(119898) ) = 119908+ [119875 (119903119886 1003816100381610038161003816119877(119898) )]120587minus (119903119886 1003816100381610038161003816119877(1) ) = 119908minus [119875 (119903119886 1003816100381610038161003816119877(1) )]120587+ (119903119886 1003816100381610038161003816119877(119887) ) = 119908+ [ 119898sum

119894=119887

119875 (119903119886 1003816100381610038161003816119877(119894) )]

minus 119908+ [ 119898sum119894=119887+1


120587minus (119903119886 1003816100381610038161003816119877(119887) ) = 119908minus [ 119887sum119894=1

119875 (119903119886 1003816100381610038161003816119877(119894) )]



(10)


119908+ (119901) = 119901120574[119901120574 + (1 minus 119901)120574]1120574

119908minus (119901) = 119901120575[119901120575 + (1 minus 119901)120575]1120575

(11)





(12)




119874119881(119877(119887)) = 119860sum119886=1


(13)




10 Complexity




1205881119894119895 11986411 11986412 119864131198771 3 5 71198772 2 3 51198773 1 2 3


4 An Example







= 4 1198631198772119860(119879+1)

= 61198631198773119860(119879+1)




Complexity 11




119875(1198641119895 1003816100381610038161003816119877119894 ) 11986411 11986412 119864131198771 01 03 061198772 02 025 0551198773 03 02 05



5 Conclusions





12 Complexity


119903119886 11986411 11986412 119864131198771 28561 16776 12250 9705 9000 42790 24794 18177 14593 12250 56194 32227 23509 18865 159161198772 9621 9000 7629 6989 6613 25557 13215 9000 7959 7371 40025 21683 14742 10664 90001198773 5878 4111 3539 3255 3086 10000 5878 4691 4111 3767 19274 7760 5878 4984 4458


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 -39953 -26594 -20357 -16735 -140431198772 -8674 -4924 -2835 -1741 -15771198773 17888 11582 8380 6591 5615


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 0 0 0 0 01198772 0541 0568 0610 0643 06341198773 1 1 1 1 1


Data Availability


Additional Points






Acknowledgments



References








Complexity 13




































14 Complexity
























Journal of












Journal of







Volume 2018






Volume 2018








10 Complexity




1205881119894119895 11986411 11986412 119864131198771 3 5 71198772 2 3 51198773 1 2 3


4 An Example







= 4 1198631198772119860(119879+1)

= 61198631198773119860(119879+1)




Complexity 11




119875(1198641119895 1003816100381610038161003816119877119894 ) 11986411 11986412 119864131198771 01 03 061198772 02 025 0551198773 03 02 05



5 Conclusions





12 Complexity


119903119886 11986411 11986412 119864131198771 28561 16776 12250 9705 9000 42790 24794 18177 14593 12250 56194 32227 23509 18865 159161198772 9621 9000 7629 6989 6613 25557 13215 9000 7959 7371 40025 21683 14742 10664 90001198773 5878 4111 3539 3255 3086 10000 5878 4691 4111 3767 19274 7760 5878 4984 4458


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 -39953 -26594 -20357 -16735 -140431198772 -8674 -4924 -2835 -1741 -15771198773 17888 11582 8380 6591 5615


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 0 0 0 0 01198772 0541 0568 0610 0643 06341198773 1 1 1 1 1


Data Availability


Additional Points






Acknowledgments



References








Complexity 13




































14 Complexity
























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 11




119875(1198641119895 1003816100381610038161003816119877119894 ) 11986411 11986412 119864131198771 01 03 061198772 02 025 0551198773 03 02 05



5 Conclusions





12 Complexity


119903119886 11986411 11986412 119864131198771 28561 16776 12250 9705 9000 42790 24794 18177 14593 12250 56194 32227 23509 18865 159161198772 9621 9000 7629 6989 6613 25557 13215 9000 7959 7371 40025 21683 14742 10664 90001198773 5878 4111 3539 3255 3086 10000 5878 4691 4111 3767 19274 7760 5878 4984 4458


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 -39953 -26594 -20357 -16735 -140431198772 -8674 -4924 -2835 -1741 -15771198773 17888 11582 8380 6591 5615


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 0 0 0 0 01198772 0541 0568 0610 0643 06341198773 1 1 1 1 1


Data Availability


Additional Points






Acknowledgments



References








Complexity 13




































14 Complexity
























Journal of












Journal of







Volume 2018






Volume 2018








12 Complexity


119903119886 11986411 11986412 119864131198771 28561 16776 12250 9705 9000 42790 24794 18177 14593 12250 56194 32227 23509 18865 159161198772 9621 9000 7629 6989 6613 25557 13215 9000 7959 7371 40025 21683 14742 10664 90001198773 5878 4111 3539 3255 3086 10000 5878 4691 4111 3767 19274 7760 5878 4984 4458


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 -39953 -26594 -20357 -16735 -140431198772 -8674 -4924 -2835 -1741 -15771198773 17888 11582 8380 6591 5615


119881(119903119886 100381610038161003816100381610038161198771198941 ) 1199031 1199032 1199033 1199034 11990351198771 0 0 0 0 01198772 0541 0568 0610 0643 06341198773 1 1 1 1 1


Data Availability


Additional Points






Acknowledgments



References








Complexity 13




































14 Complexity
























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 13




































14 Complexity
























Journal of












Journal of







Volume 2018






Volume 2018








14 Complexity
























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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