Research ArticleA Public-Participation-Based Mixed MultiattributeDecision-Making Approach for Major Public Affairs

Chenguang Cai 1 Yong Luo 2 Guiju Zhu 3 and Hao Zou 4

1School of Accounting Hunan University of Finance and Economics Changsha 410205 China2School of Economics and Management Sichuan Tourism University Chengdu 610100 China3School of Management Hunan University of Technology and Business Changsha 410205 China4School of Business Administration Hunan University of Finance and Economics Changsha 410205 China

Correspondence should be addressed to Yong Luo luoyongcds163com

Received 14 May 2021 Revised 7 July 2021 Accepted 22 July 2021 Published 9 August 2021

Academic Editor Ewa Rak

Copyright copy 2021ChenguangCai et alis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

e decision-making activities of major public affairs are closely related to the public so the decision results of such affairs must besupported by the public e public must participate in decision-making activities to ensure their effectiveness which furtherincreases the complexity In addition attribute information and public opinion usually present different forms of expression inthis type of problem making decision-making more difficult erefore a suitable decision approach must be chosen to deal withthis type of decision problem is paper addresses the decision-making characteristics of major public affairs and proposes apublic-participation-based decision-making approach for mixed multiattribute decision-making problems in major public affairse proposed approach can work with entirely unknown attribute weights and decision-making values represented in multipleformats First the statistical distribution of public opinions is determined based on the expectation of the attributes resulting indecision-making reference points for various attributes e different forms of attributes and reference points are unified enthe values of the attributes and reference points are standardized Afterward the attribute prospect value for each alternative iscalculated using the attribute value and corresponding reference pointse attribute weight intervals are determined based on theimportance information of the attributes provided by the public An optimization model is established to determine the attributeweights to maximize the alternative attribute deviation Next the comprehensive prospect value of each alternative is obtained todetermine the ranking of the alternatives Finally a case analysis is conducted with a method comparison and sensitivity analysisand the feasibility and effectiveness of the proposed approach are verified In the proposed method the reference points for eachattribute are set according to the distribution characteristics and ambiguity of public expectations guaranteeing that publicexpectations can be effectively reflected in the attribute reference points In the process of attribute weighting based on theinformation for the attribute importance given by the public the range of attribute weights is determined en we obtain theexact value of the attribute weights using an optimization model to maximize the alternative attribute deviatione final result ofthe attribute weights ensures the full expression of public opinion and can improve the differentiation of decision results which isconvenient for ranking alternatives During evaluation of the alternatives based on prospect theory the expression forms ofattributes and reference points are unified Subsequently the values of them are normalized which satisfy the decision-makingrequirement of major public affairs

1 Introduction

Major public affairs have a profound influence and areclosely related to the vital interests of the public Conse-quently they attract a considerable amount of societal at-tention Once the public does not support the final decision

results it is easy to generate social risks leading to a series ofsocial problems [1 2] Numerous public opinions must begathered across different strata of society to ensure that thedecision-making results regarding major public affairs fullyreflect the majority public opinion determining the decisionoutcomes [3 4] In the process of decision-making given the

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 7550055 14 pageshttpsdoiorg10115520217550055

variations in individual opinions on major public affairs thedistribution of public opinions presents a distinct charac-teristic of discreteness In addition the scale of publicopinion is usually immense which makes it challenging tocollect process and analyze public opinions us effectivedecision-making on major public affairs based on collectingand analyzing public opinions is an important concern forgovernmental authorities

e department of public affairs management usesvarious means to encourage the public to participate indecision-making to ensure that information collected fromthe public is effective and extensive Popular public-par-ticipation methods include questionnaire surveys onlineplatform messages and on-site hearings [5ndash8] Some re-searchers have introduced statistical theory for sorting andanalyzing public opinions identifying representative publicopinions based on their statistical distribution characteris-tics and providing tools for identifying and processingpublic opinions [9] Most citizens lack professionalknowledge making it difficult for them to provide objectiveguidance and suggestions when contributing to decision-making e opinions they provide focus more onexpressing their personal expectations or concerns Ingeneral the stronger the public expectations or appeals forcertain aspects of a topic the more the attention this aspectneeds to receive [10ndash12] erefore the public expectationsshould be included in the decision-making of public affairsas an important reference standard for alternative evalua-tion It is crucial to set the decision-making reference pointseffectively based on public expectations Previous researchon decision-making reference points has focused on theeffectiveness evaluation of setting reference points [13 14]the distance measurement between the reference points andalternative attributes [15] and the dynamic evolutioncharacteristics of the reference points [16] Most existingmethods for setting reference points are based on knownindividual data or a small amount of known data How to setthe reference points based on large amount of publicopinions requires further study

For the decision problems the results of attribute weighthave a great influence on decision-making so choosing aproper weighting method is necessary for decision-makingactivities Common weighting methods include analytichierarchy process (AHP) [17] entropy weight [18] andmaximum deviation methods [19] In an actual weightingoperation process the attribute weighting methods of de-cision-making problems are primarily chosen based onseveral factors the decision-making problem type attributedata expression type distribution characteristics of the at-tribute data and others [20 21] For decision-making inpublic affairs in order to ensure the effectiveness of attributeweighting the public is introduced to decision-makingactivities Public opinions on the importance of the attri-butes must be considered when determining attributeweights to ensure the effectiveness of attribute weightingHowever the existing attribute weighting approaches areseldom involved in public opinion erefore further re-search is required on how to effectively weight the attributesconsidering public opinions

Different attributes reflect the various contents of eachalternative hence according to the actual needs of the at-tribute expression each attribute is usually expressed in adifferent form such as a crisp value interval value andlinguistic value For a decision problem if the attributes areexpressed in different forms we call this decision problem asmixed multiattribute decision problem [22ndash24] Publicdecision-making activities are usually complicated in orderto ensure the effectiveness of the decision different types ofattributes are introduced to explain the decision alternativesus public decision-making activities usually have thecharacteristics of mixed multiattribute decision-makingproblems the mixed multiattribute characteristics of deci-sion-making problems must be considered when dealingwith public decision problems

In the research regarding decision-making methodssome scholars have introduced fuzzy number operators todecision-making activities which can provide methodicalsupport for the operation of decision-making information[25ndash29] Moreover some decision methods such as theTechnique for Order of Preference by Similarity to IdealSolution (TOPSIS) [30] VIekriterijumsko KOmpromisnoRangiranje (VIKOR) method [31] Elimination and ChoiceExpressing Reality (ELECTRE) method [32 33] and theprospect theory [34] are widely used in various types ofdecision problems Some scholars have used social networkanalysis to rank schemes based on the relationship and trustbetween decision-makers [35 36] ese research achieve-ments provide necessary technical and methodologicalsupport for solving various kinds of decision-makingproblems

Major public affairsrsquo decision-making has a high degreeof complexity and uncertainty To ensure decision-makingeffectiveness we must consider the actual characteristics ofthe decision-making problems before the decision method isdetermined

According to the above the existing research can providenecessary references for the study of major public affairsHowever the decision-making problem of major publicaffairs requires public participation therefore the charac-teristics of public opinions must be considered in the aspectsof attribute reference point setting attribute weighting andalternative ranking Relatively few research achievementsexist on public-participation decision-making problems sofurther research is needed regarding this aspect Based on theabove analysis a mixed multiattribute decision-makingapproach with public participation is proposed in this paperCompared to the existing research the contribution of thispaper is reflected in the three main aspects

First we set attribute reference points based on publicexpectations Public expectations have a large volume andhigh dispersion characteristics which must be collated andanalyzed In addition public expectations are subjective anduncertain making it more challenging to deal with thisinformation Existing research results rarely consider publicexpectations when setting reference points so the existingreference point setting methods are unsuitable for this de-cision-making background Based on the above analysis wepropose a reference point setting method considering public

2 Mathematical Problems in Engineering

expectations We determine the distribution and ambiguityof public expectation according to the expectation infor-mation given by the public and on this basis the referencepoint for each attribute is set

Second we propose an attribute weighting methodconsidering public opinions As we know the value of at-tribute weights has a direct effect on the outcomes of de-cision-making Existing attribute weighting methods rarelyconsider public opinion during the decision-making processin major public affairs To ensure that attribute weightingresult is acceptable to the public we propose an attributeweighting method considering public opinions In this at-tribute weighting method we determine the value range ofattribute weights according to public opinion and establishthe attribute weighting optimization model to determine thefinal attribute weights e result of the attribute weightsobtained using this method reflects the public opinion andguarantees the validity of weighting

ird we introduce the decision-making method of theprospect theory to rank the alternatives in public-partici-pation scenarios e prospect theory considers the un-certainty of events and the decision risk and explains thestructural effect preference nonlinearity resource depen-dence risk pursuit and loss avoidance reflected in peoplersquoschoices [37]erefore the prospect theory is suitable for thedecision-making needs of major public affairs However theexisting research on the prospect theory is not involved inpublic-participation scenarios us the existing decision-making method of prospect theory must be improvedaccording to the characteristics of public participation inmajor public affairs to make it suitable for this type ofdecision-making problem Based on the above analysis weintroduce an improved decision-making method of theprospect theory which requires taking the expectation ofpublic groups as the reference point to guarantee that thedecision-making results better reflect public opinion In thedecision-making process the expression forms of attributeinformation and public information vary to ensure theimplementation of decision-making Different expressionforms of attribute information and public opinions must beunified into an interval before the normalized operation

e rest of the paper is organized as follows In Section 2the preliminaries summarize the knowledge that forms thebasis of the paper and in Section 3 themethod and principleare elucidated explaining the proposed method Section 4presents the case analysis to verify the rationality and validityof the approach In Section 5 a comparison of methods andsensitivity analysis are discussed Finally Section 6 providesthe conclusions

2 Preliminaries

Definition 1 (see [38]) Given a linguistic set S s01113864

s1 sT s0 and sT are the lower and upper limits of thelinguistic variables respectivelye conversion relationshipbetween the linguistic variables st (t 0 1 2 T) andinterval numbers at [aL

t aUt ] is as follows where

0le aLt le aU

t le 1

aL0 0

aUt a

Lt +

1T

0le tleT

aLt a

Utminus 1 1le tleT

aUT 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

Definition 2 Given the uncertain linguistic variables [st1

st2] st1

st2isin S where S s0 s1 sT1113864 1113865

(t1 t2 0 1 2 T 0le t1 le t2 leT) e conversion rela-tionship between the uncertain linguistic variables [st1

st2] and interval number at [aL

t aUt ] (0le aL

t le aUt le

1) is as follows

aLt

t1

T 0le t1 leT

aUt

t2

T 0le t2 leT

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(2)

Definition 3 (see [39]) Suppose that the interval numbera [aL aU] is a nonnegative interval number (ie0le aL le aU) then the midpoint of a is defined bya (aL + aU)2 and the interval number radius of a isdefined by ca (aU minus aL)2

3 Method and Principle

31 Problem Description e decision-making problems inthis study satisfy the following basic assumptions

(1) e set of alternatives is certain(2) e set of attributes to describe the alternatives is

certain(3) e expression form of each attribute is known(4) e public individuals provide expectations based on

their psychological perceptions and the form ofexpression of public expectations is consistent withthe form of expression of the attributes

(5) e public provides the attribute evaluation value ofimportance in the form of linguistic or uncertainlinguistic variables

Suppose that the set of decision-making alternatives for amajor public affair is Z z1 z2 zM1113864 1113865 with attribute setG g1 g2 gN1113864 1113865 and attribute weightω (ω1ω2 ωN)T 1113936

Nj1 ωj 1 e value of attribute j

in alternative i is yij i 1 2 M j 1 2 N More-

over yij can be expressed in the form of crisp numbers

interval numbers linguistic variables or uncertain linguisticvariables e public provides expectation values andevaluates the importance values of the attributes as per theirpreferences Suppose that the number of individuals par-ticipating in the evaluation of the expected value of attribute

Mathematical Problems in Engineering 3

j is Hj e expected value given by individual k on attributej is rk

j where k 1 2 Hj which can be in crispnumbers interval numbers linguistic variables or uncertainlinguistic variables Suppose that the number of public in-dividuals participating in evaluating the importance of at-tribute j is ζj e evaluation value of importance providedby public individual k on attribute j is bk

j wherek 1 2 ζj which can be in linguistic or uncertainlinguistic variables depending on the publicrsquos actual need forattribute expression

e study problem is determining the reference point foreach attribute and the attribute weights according to theattribute values for each alternative the public expectedvalues for each attribute and the public-evaluated impor-tance value for each attribute In addition this study assesseshow to choose a practical decision-making approach to rankall alternatives to determine the optimal alternative

32 Determination of Attribute Reference Points Based onPublic Opinions e expression form of public expectedvalues is primarily related to two factors the attribute ex-pression form of the alternative and the accuracy of indi-vidual expected values If the attribute value of thealternative is expressed in real numbers (crisp numbersinterval numbers etc) the public expected value of theattribute is also expressed in real numbers (crisp numbersinterval numbers etc) In contrast if the attribute value ofthe alternative is expressed in linguistic or uncertain lin-guistic variables the public expected value of the attribute isalso expressed in linguistic or uncertain linguistic variables

e other factor is the accuracy of the individual ex-pected values When the public provides their expectedvalues of an attribute some of the public can express theirexpected values accurately Consequently this portion of thepublic chooses to provide their expected value of the at-tribute in crisp numbers or linguistic variables In contrastthe other portion of the public is affected by internal orexternal factors and cannot accurately express their expectedvalues for the attributes is portion of the public usuallyexpresses their expected values in interval numbers or un-certain linguistic variables

According to the above analysis the expected values ofattributes given by the public also have various forms ofexpression and thus are expressed in different forms eexpression forms must be normalized to determine thespecific distribution and ambiguity of the public expectedvalues and obtain the reference point for each attribute ereference points for attributes can be determined as follows

(1) e initial public expected values of the attribute areprocessed which is achieved as follows

Step 1 e public expected values for the attributefrom linguistic or uncertain linguistic vari-ables are converted into an interval valueusing Definitions 1 and 2

Step 2 e converted public expected values areexpressed in crisp or interval numbersAccording to Definition 3 if the public

individualrsquos expected value rkj is an interval

number described as rkj [rkL

j rkUj ] its

ambiguity is ckj (rkU

j minus rkLj )2 If rk

j is a crispnumber its ambiguity is ck

j 0 e com-prehensive expected ambiguity is defined asthe average expected ambiguity of all publicindividuals for attribute j (iecj (1Hj) 1113936

Hj

k1 ckj)

Step 3 Definition 3 converts the interval form of thepublic individual expected values rk

j intocrisp numbers denoted by rk

j rk

j (rkLj + rkU

j )2 e crisp form of publicexpected values rk

j retains the original formwhich does not need to be converted Afterthe conversion operation the expectationgiven by individual k on attribute j is a crispvalue defined as rk

j

(2) Based on the distribution of rkj the probability

distribution of the expected values on different at-tributes can be determined According to a previousstudy [40] large-scale public opinions usually followa normal distribution

(3) e attribute reference points are determined

Step 1 Under the normal distribution situation forrk

j the mean of public expectation distri-bution over rk

j is defined as μ(rkj ) which can

be determined based on the distribution ofrk

j e distribution variance of the publicexpected value on attribute j is defined asσ(rk

j )Step 2 e attribute reference point 1113957rlowastj is confirmed

which is expressed in the form of intervalnumbers1113957rlowastj [1113957rlowastLj 1113957rlowastUj ] [μ(rk

j ) minus cj μ(rkj) + cj] e

attribute reference point 1113957rlowastj can be deter-mined using the mean of public expectationdistribution μ(rk

j) and the comprehensiveambiguity of the public expected values cj

33 Calculation of the Alternative Prospect Value

331 Normalization of the Attribute Values and ReferencePoints According to Section 32 the finalized form of theattribute reference point is an interval number whereas theattribute value of the alternative can be expressed as a crispnumber interval number linguistic variable or uncertainlinguistic variable e dimensions of various attributes areinconsistent so the attribute values and reference points ofall alternatives must be normalized

e attribute values are unified in the form of intervalnumbers If the attribute value of the alternative is a crispnumber it is rewritten as an interval number with equalupper and lower limits If the attribute value of the al-ternative is a linguistic or uncertain linguistic variablethen Definitions 1 and 2 can convert it into an intervalnumber e interval form of yi

j is defined asyi

j [yiLj yiU

j ]

4 Mathematical Problems in Engineering

Next the attribute values and attribute reference pointsare normalized Equations (3)ndash(6) are used to normalize theattribute values and reference points in interval numbers toeliminate the dimension influence of the original data e

normalized attribute value of yij is

pij [pLij pU

ij] 0lepLij lepU

ij le 1 and the normalized attributereference point of 1113957rlowastj is qj [qL

j qUj ] 0le qL

j le qUj le 1

Attribute gj is a profit index

pLij

yiLj minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

pUij

yiUj minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(3)

qLj

1113957rlowastLj minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

qUj

1113957rlowastUj minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(4)

In addition attribute gj is a cost index

pLij

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus y

iUj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

pUij

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus y

iLj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

qLj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus 1113957r

lowastUj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

qUj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus 1113957r

lowastLj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(6)

332 Calculation of Prospect Profit and Loss Value overDifferent Attributes e profit Gij and loss Fij of attributevalue pij are calculated according to the relationship be-tween the normalized attribute value pij and the nor-malized reference point qj e equations to calculate Gij

and Fij are presented in Table 1 e values of v(+)ij and v

(minus )ij

are determined based on the prospect theory as given inequation (7) According to a previous study [41] the co-efficients of α β λ are α β 088 and λ 225

v(+)ij Gij1113872 1113873

α Gij ge 0

v(minus )ij minus λ minus Fij1113872 1113873

β Fij lt 0

⎧⎪⎨

⎪⎩(7)

e prospect profit-loss matrix of the attribute can beconstructed as follows VM [vij]MtimesN where the value ofvij can be obtained using the following equation

vij v(+)ij + v

(minus )ij (8)

34 Determination of Attribute Weights First the publicevaluation information of the attribute importance is pro-cessed Public individuals give their evaluation values of theimportance of different attributes in the form of linguistic oruncertain linguistic variables Some individuals can expresstheir opinions more accurately so they choose to evaluatethem in linguistic variables Others feel a certain degree ofambiguity or uncertainty over the evaluation results of their

Mathematical Problems in Engineering 5

given attribute importance so they choose to express theiropinions using uncertain linguistic variables According tothe expression characteristics of the public Definitions 1 and2 can be used to convert the public evaluation values fromlinguistic or uncertain linguistic variables into intervalnumbers If the number of individuals participating in theimportance evaluation of attribute j is ζj the importanceevaluation value given by individual k over attribute j isεk

j k 1 2 ζj Based on Definitions 1 and 2 εkj can be

converted into the interval type bkj [bkL

j bkUj ]

k 1 2 ζj We take the average value of the importanceevaluation of the public on attribute j as the comprehensiveimportance evaluation value of attribute j (ie bj [bL

j bUj ]

where bLj (1ζj) 1113936

ζj

k1 bkLk bU

j (1ζj) 1113936ζj

k1 bkUk 0le bL

j lebU

j le 1)Second the value range of the attribute weights ωj is

determined e value range of the attribute weights ωj isassumed to be ωj isin [ωL

j ωUj ] Based on the public com-

prehensive evaluation value of the importance over attributej the upper and lower limits of ωj are determined

ωLj

bLj

bLj + 1113936

Nminus 1e1enejb

Ue

ωUj

bUj

bUj + 1113936

Nminus 1e1enejb

Le

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)

Theorem 1 For ωj isin [ωLj ωU

j ] ωj must exist that meets theconstraints of 0leωj le 1 and 1113936

Nj1 ωj 1

Proof Because 0le bLj le (bL

j + 1113936Nminus 1e1enejb

Ue ) and

0le bUj le (bU

j + 1113936Nminus 1e1enejb

Le ) 0le (bL

j (bLj + 1113936

Nminus 1e1enejb

Ue ))le 1 and

0le (bUj (bU

j + 1113936Nminus 1e1enejb

Le ))le 1 us it can be deduced that

0leωj le 1 As 0le bLj le bU

j le 1 and ωLj (bL

j (bLj + 1113936

Nminus 1e1enej

bUe ))le (bL

j 1113936Nj1 bL

j ) and it can be deduced that (1113936Nj1 ω

Lj

1113936Nj1[bL

j (bLj + 1113936

Nminus 1e1enejb

Ue )])le 1113936

Nj1(bL

j 1113936Nj1 bL

j ) 1 Simi-larly it is deduced that 1113936

Nj1 ω

Uj ge 1 As the value of ωj is

continuous within the range [ωLj ωU

j ] and 0leωLj leωU

j le 1 so1113936

Nj1 ωj 1 must existird we determine the attribute weights An optimi-

zation model is constructed to solve the attribute weights tomaximize the dispersion of attributes on all alternatives

maxψ ωj1113872 1113873 1

M1113944

M

i11113944

N

j1

pLij minus 1113957p

Lj

11138681113868111386811138681113868

11138681113868111386811138681113868 + pUij minus 1113957p

Uj

11138681113868111386811138681113868

11138681113868111386811138681113868

2⎛⎝ ⎞⎠ωj

st

ωLj leωj leω

Uj

1113944

N

j1ωj 1

1113957pj 1113957pLj 1113957p

Uj1113960 1113961

1M

1113944

M

i1p

Lij

1M

1113944

M

i1p

Uij

⎡⎣ ⎤⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)

Table 1 Equations to calculate Gij and Fij

No e relationship between pij and qj e loss Fij e profit Gij

1 pUij lt qL

j 05(pLij + pU

ij) minus qLj 0

2 qUj ltpL

ij 0 05(pLij + pU

ij) minus qUj

3 pLij lt qL

j lepUij lt qU

j 05(pLij minus qL

j ) 0

4 qLj ltpL

ij le qUj ltpU

ij 0 05(pUij minus qU

j )

5 pLij lt qL

j lt qUj ltpU

ij 05(pLij minus qL

j ) 05(pUij minus qU

j )

6 qLj lepL

ij ltpUij le qU

j 0 0

6 Mathematical Problems in Engineering

Theorem 2 Model (10) must have an optimal solution

Proof Under the constraint ωj isin [ωLj ωU

j ] there must be areasonable value of ωj that satisfies 1113936

Nj1 ωj 1 so the

feasible domain of the attribute weight is a nonempty setIn addition 0lepij le 1 and 0lepj le 1 so it is easy to de-duce that 0le ((|pL

ij minus pLj | + |pU

ij minus pUj |)2)le 1 As known

0leωj le 1 1113936Nj1 ωj 1 thus it can be deduced

that 0le 1113936Nj1((|pL

ij minus pLj | + |pU

ij minus pUj |)2)ωj le 1 and 0le

ψ(ωj) (1M) 1113936Mi1 1113936

Nj1 ((|pL

ij minus 1113957pLj |+ |pU

ij minus 1113957pUj |)2)ωj le 1

As ψ(ωj) is a bounded continuous function the constraintcondition of the attribute weight is a bounded closed set soModel (10) must have an optimal solution

To sum up the specific steps of the proposed decision-making approach are as follows

Step 1 e public expected values of attributes withvarious expressions are converted into crisp numbersStep 2e distribution of the public expected values isdetermined on all attributes according to the publicexpected opinion in the crisp number type Next theattribute reference points are obtained based on thedistribution mean of the public expectation andcomprehensive ambiguity of public expected valuesStep 3 e attribute value and reference points arenormalized e attribute prospect value of each al-ternative is calculated based on the prospect theoryStep 4 e value range of the attribute weights isdetermined using equation (9) and Model (10) to de-termine the attribute weightStep 5 e comprehensive prospect values of differentalternatives are obtained using equation (11) to realizethe ranking of alternative alternatives

Vi 1113944N

j1ωjvij (11)

4 Case Analysis

We take a subway construction project as an example toverify the rationality and effectiveness of the method pro-posed in this paper A provincial capital city plans to extendthe No 2 subway line to the west for which three alter-natives can be selected e extension of the subway line willmake public transportation more convenient for residentsalong the line However the subway construction will take along time to be completed occupy a large amount of publicspace and generate substantial dust which interferes withthe daily life of the surrounding residents According to theconstruction requirements and characteristics of the subwayline the organizers of the decision-making activity selectedfour attributes to evaluate the alternatives the averagedistance between the stations and densely populated areasalong the line (g1 units m cost-based index) estimated

construction time (g2 units month cost-based index)enclosed public area for construction (g3 units m

2 cost-based index) and dust and sand treatment effect (g4qualitative index profit-based index) Among these alter-natives g1 is expressed in crisp numbers g2 and g3 areexpressed in interval numbers and g4 is expressed in lin-guistic or uncertain linguistic variables e conversionstandard between the dust and sand treatment effect andlinguistic variables is presented in Table 2 and the attributevalues of different alternatives are listed in Table 3

e primary public group affected by the constructionand operation of the subway is urban residents ereforeduring the decision-making process the opinions of thepublic group directly affected by the subway must be fullyconsidered Various media-driven methods were used topublicize the project to enable the public to understand theactual subway project situation better e public couldexpress their opinions on the subway project using differentmethods such as online platforms telephone and mailquestionnaires e public provides two pieces of evaluationinformation based on their opinions of the attributes theirexpectations and the importance evaluation value econversion standard between the evaluation values of theimportance and linguistic variables is given in Table 2

When the public opinion survey was finished the or-ganizers of the public opinion survey identified and countedthe public individuals who effectively participated estatistical results of the public expectations are listed inTable 4 and the statistical results of the public importanceevaluation values of different attributes are presented inTable 5 Numerous individuals participated in the surveyeffectively Due to the space limitations of the article we onlypresent the partial statistical results of the public opinions inTables 4 and 5

e original public expected values of the attributes inTable 4 were processed First the public expected values ofthe attributes were converted into interval numbers Nextthe comprehensive ambiguity of the expected values of theattributes was calculated which was shown in Table 4 enthe public expected values of the attributes in intervalnumbers were converted into crisp numbers in Definition 3the details of the public expected values for different attri-butes in the form of crisp numbers are shown in Table 6

Finally based on the relevant content in Table 6 thedistribution of public expected values for various attributeswas examined e distribution of public expected values onvarious attributes was examined e fittings of the distri-butions are illustrated in Figures 1ndash4 According to thestatistical distribution results and comprehensive ambiguityof the public expected values the reference point of eachattribute was determined as presented in Table 7

Equations (3)ndash(6) were used to normalize the attributereference points and attribute values of different alternativese normalized attribute reference points and alternativeattribute values are listed in Table 8

Next equations (7) and (8) were used to calculate theprospect values of the attributes and the prospect profit-lossmatrix of the attributes is expressed

Mathematical Problems in Engineering 7

Table 2 Conversion standards between dust and sand treatment (attribute importance) and linguistic variables

Dust and sand treatmenteffect (attribute importance)

Extremely poor (canbe ignored)

Terribly poor(extremely

unimportant)

Very poor (veryunimportant) Poor (unimportant) Fair

(good)

Linguistic variable s0 s1 s2 s3 s4

Dust and sand treatmenteffect (attribute importance) Good (important) Very good (very

important)Extremely good

(extremely important)Perfect (maximum

importance) mdash

Linguistic variable s5 s6 s7 s8 mdash

Table 3 Attribute values of various alternatives

AlternativesAverage distance between the station

and the population gatheringarea along the line (m)

Estimated constructiontime of the

project (months)

Construction enclosedpublic areas (m2)

Dust and sandtreatment

effect (qualitative index)1 330 [64 71] [28732 29849] [S6 S7]2 379 [61 65] [25373 27711] S73 336 [64 68] [28064 29849] [S6 S7]

Table 4 Statistics of public expected values for different attributes

Average distance between thestation and the population

gathering area along the line (m)

Estimated construction time ofthe project (months)

Construction-enclosed publicareas (m2)

Dust and sand treatment effect(qualitative index)

Number of effective participants Number of effective participants Number of effective participants Number of effective participants5472 6070 5518 5491

No Expectedvalues Ambiguity No Expected

value Ambiguity No Expectedvalue Ambiguity No Expected

value Ambiguity

1 [453 499] 23 1 [55 58] 15 1 [3052830789] 1305 1 [S4 S5] 00625

2 [377 377] 0 2 [64 66] 1 2 [2621229284] 1536 2 [S5 S5] 0

5471 [346 352] 3 6069 [67 74] 35 5517 [3004933846] 18985 5490 [S5 S6] 00625

5472 [338 374] 18 6070 [70 78] 4 5518 [2817531285] 1555 5491 [S6 S6] 0

Table 5 Public evaluation results of the importance for different attributes

Average distance betweenthe station and the

population gathering areaalong the line (m)

Estimated constructiontime of the project

(months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

Number of effectiveparticipants

Number of effectiveparticipants

Number of effectiveparticipants

Number of effectiveparticipants

4186 4182 4186 4185No Importance No Importance No Importance No Importance1 [S3 S4] 1 [S1 S2] 1 [S3 S3] 1 [S3 S5]2 [S3 S4] 2 [S1 S2] 2 [S3 S3] 2 [S3 S5]

4185 [S3 S6] 4181 [S2 S2] 4185 [S1 S4] 4184 [S4 S5]4186 [S3 S6] 4182 [S0 S1] 4186 [S1 S4] 4185 [S4 S5]

8 Mathematical Problems in Engineering

Table 6 e public expected values for different attributes in the form of crisp numbers

Average distance betweenthe station and the

population gathering areaalong the line (m)

Estimated construction timeof the project (months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

No Expected value No Expected value No Expected value No Expected value1 476 1 56 1 306585 1 056252 377 2 65 2 27748 2 05000

5471 349 6069 705 5517 319475 5490 068755472 356 6070 74 5518 29730 5491 07500

250 300 350 400 450 500 550 600 6500

20

40

60

80

100

120

140

160

180

Freq

uenc

y

Frequency

Public expectation of attribute 1 (m)

Normal distribution fitting curve

Figure 1 Fitting curve for public expected values of attribute 1

55 60 65 70 75 80 85 900

50

100

150

200

250

300

350

Freq

uenc

y

FrequencyNormal distribution fitting curve

Public expectation of attribute 2 (months)

Figure 2 Fitting curve for public expected values of attribute 2

22 24 26 28 3 32 34 36 380

20

40

60

80

100

120

140

Freq

uenc

y

Frequency

times104

Normal distribution fitting curve

Public expectation of attribute 3 (m2)

Figure 3 Fitting curve for public expected values of attribute 3

05 06 07 08 09 1 11 12Public expectation of attribute 4 (qualitative index)

0

500

1000

1500

2000

2500

Freq

uenc

y

FrequencyNormal distribution fitting curve

Figure 4 Fitting curve for public expected values of attribute 4

Mathematical Problems in Engineering 9

VM

08287 02070 00000 02164

04719 04752 03587 03983

07864 02070 00000 02164

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (12)

e value range of the attribute weights can be deter-mined using equation (9) based on the conversion betweenlinguistic variables and interval numbers ω1 isin [024 055]

ω2 isin [007 026] ω3 isin [010 034]ω4 isin [017 044] Attri-bute weights can be determined using Model (10)ω1 042ω2 007ω3 034ω4 017

Based on the attribute prospect value and attributeweights the comprehensive prospect value of each alter-native can be obtained using equation (11) whereV1 03994 V2 04211 and V3 03816 e ranking ofthree alternatives is V2 gtV1 gtV3 erefore V2 is the op-timal alternative

5 Comparison of Methods andSensitivity Analysis

51ComparisonofMethods To verify the effectiveness of theproposed approach we introduce two existing decision-making methods e first method is the double-referencepoint decision-making method based on the prospect theory[42] e second method is the TOPSIS method [43] In thetwo mentioned methods positive and negative ideal pointsare set as reference points used as the basis of the alternativeevaluation e positive and negative reference points ofattribute j are defined as 1113957rlowast+j and 1113957rlowastminusj respectively Based onthe prospect theory and TOPSIS method the comparisonbetween the attribute value of the alternative and the cor-responding reference points is analyzed and the evaluationvalue of each alternative is obtained In the double-referencepoint decision-making method based on the prospect

theory the maximum and minimum values of public ex-pectation are taken as positive and negative reference points

Public expectations obey the normal distributiontherefore the maximum and minimum values of publicexpectations can be obtained using the three-sigma (3σ)

theorem of normal distribution [44] For attribute j themaximum and minimum values of public expectations areobtained using the 3σ theorem of normal distributiondenoted by μ(rk

j) + 3σ(rkj) and μ(rk

j) minus 3σ(rkj) respectively

According to Definitions 1 and 2 the conversion values oflinguistic variables or uncertain linguistic variables are in therange [0 1] In the conversion crisp valuesrsquo distribution oflinguistic variables or uncertain linguistic variables in orderto ensure the maximum and minimum values we meet therange requirements in this type of distribution we define themaximum and minimum values as follows ifμ(rk

j ) + 3σ(rkj )gt 1 we set maximum value as 1 if

μ(rkj ) minus 3σ(rk

j )lt 0 we set minimum value as 0 For the otherscenarios the maximum and minimum values are set asμ(rk

j ) + 3σ(rkj ) and μ(rk

j) minus 3σ(rkj ) To satisfy the calculation

need of prospect value we set the positive and negativereference points of the attribute as interval numbers withequal upper and lower limits expressed as 1113957rlowast+j [(μ(rk

j) +

3σ(rkj)) (μ(rk

j) + 3σ(rkj))] and 1113957rlowastminusj [(μ(rk

j ) minus 3σ(rkj))

(μ(rkj ) minus 3σ(rk

j ))] In the TOPSISmethod we set the positiveand negative ideal points according to the maximum andminimum values of each attribute in all alternatives Forexample the positive and negative reference points of at-tribute j are respectively expressed as1113957rlowast+j [maxi12MyiU

j maxi12MyiUj ] and

1113957rlowastminusj [mini12MyiLj mini12MyiL

j ] e alternativeranking results of different decision methods are listed inTable 9

According to Table 9 the alternative ranking resultscorresponding to the existing methods proposed are

Table 8 Normalized attribute reference points and attribute values for different alternatives

AlternativeAverage distance between the stationand the population gathering area

along the line (m)

Estimated constructiontime of the project

(months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

1 [10000 10000] [00458 07137] [00318 02734] [03243 10000]2 [06182 06182] [06183 10000] [04943 10000] [10000 10000]3 [09532 09532] [03321 07137] [00318 04179] [03243 10000]Attributereference points [00000 01922] [00000 03798] [00000 04353] [00000 06486]

Table 7 Results of public expectation reference points for different attributes

AttributesAverage distance between the stationand the population gathering area

along the line (m)

Estimatedconstruction time ofthe project (months)

Construction-enclosedpublic areas (m2)

Dust and sandtreatmenteffect

(qualitative index)

Statistical distribution Normal distribution N (4460 74412) Normal distributionN (6949 8482)

Normal distribution N(28990 28992)

Normal distributionN (075 0152)

Comprehensive ambiguityof the expectations 1234 199 100621 006

Attribute reference point [43367 45834] [6750 7148] [2798379 2999621] [069 081]

10 Mathematical Problems in Engineering

inconsistent with the results obtained in this studyemainreason for the inconsistency of decision results is that thereference point setting in each decision method is different

e setting principle of reference points for the twoexisting methods is that reference points are obtainedaccording to the public expectation or the attribute value of thealternative e setting principle for the reference points usedin the two existing methods is relatively simple but it does notconsider the characteristics of public expectations For deci-sion-making activities in major public affairs public expec-tations must be fully considered to ensure the effectiveness ofthe decision-making results erefore public expectationsmust be considered when setting reference points

Based on the above analysis we develop the settingprinciple for reference points by considering public ex-pectations First the information on public expectations isdiscrete and has the characteristics of a normal distributionthus the mean value of public expectations is taken as thestandard setting for the reference point so that the referencepoint can effectively reflect the expectations of the publicgroup Second due to the uncertainty of the expectationevaluation by public individuals the expected value isusually expressed in an interval number which reflects theambiguity of public expectations e comprehensive ex-pected ambiguity of the public is used to represent theuncertain characteristics of public expectations e refer-ence point is set by the mean value of public expectationsand the comprehensive expected ambiguity of the public tobetter reflect the actual situation of public expectation ex-pression e results of the reference points are presented asinterval numbers e value of the reference points reflectsthe expectations of the public group and considers theuncertainty of the public evaluation

If an attribute value in interval form falls within therange of the corresponding reference points in whole or inpart we consider that part of the attribute value that fallswithin the range of the reference points exactly meetspublic expectations which means that the prospect valuefor this part of the attribute is zero erefore when theprospect value is calculated based on the interval reference

point the inclusion or cross relationship between theattribute interval and corresponding reference point in-terval should be considered For example Table 8 indicatesthat the normalized values of attribute 3 for alternatives 1and 3 are p13 [00318 02734] and p33 [00318

04179] respectively Moreover p13 and p33 are not equaland the normalized reference point for attribute 3 isq3 [00000 04353] It is apparent that p13 and p33 areentirely within the interval range of q3 We can affirm thatthe prospect values of attribute 3 for alternatives 1 and 3are both zero (v13 v33 0) In addition we use equations(7) and (8) and Table 8 to calculate the prospect values ofp13 and p33 we can also obtain the same result If wechoose the double-reference point decision-makingmethod to calculate the prospect value of p13 and p33 theresults of the prospect value for p13 and p33 arev13 minus 07329 and v33 minus 06732 which is different fromthe result of the proposed method erefore a differentsetting principle for the reference point leads to differentdecision-making results explaining why the sorting resultsare different in Table 9 erefore if we need to set ref-erence points for decision-making we must set themaccording to the actual situation for the decision problemand decision requirements to guarantee the effectiveness ofthe decision results

52 Sensitivity Analysis of the Reference Point Interval RangeAccording to the content above the comprehensive ambi-guity of public expectations is related to the interval range ofattribute reference points affecting the decision-makingresultse interval adjustment coefficient of reference pointθ (0le θle 1) is introduced in the expression of referencepoints to study the relationship between the value range ofthe reference points and decision results further e ref-erence point of attribute j is defined as1113957rlowastj [1113957rlowastLj 1113957rlowastUj ] [μ(rk

j ) minus θcj μ(rkj) + θcj] e compre-

hensive ambiguity of public expectations for each attribute cj

is known When the values of θ are 0 025 05 075 and 1the corresponding value range of reference points changes

Table 9 Alternative ranking results of different decision methods

Methods Attributereference points Decision results Ranking of

the alternatives

Double-reference point decision-makingmethod based on prospect theory

1113957rlowast+1 [22277 22277] 1113957r

lowast+2 [4405 4405]

1113957rlowast+3 [2029300 2029300] 1113957r

lowast+4 [100 100] V1 minus 01738

V2 minus 00983

V3 minus 01654

V2 gtV3 gtV11113957rlowastminus1 [66923 66923] 1113957r

lowastminus2 [9493 9493]

1113957rlowastminus3 [3768700 3768700] 1113957r

lowastminus4 [000 000]

TOPSIS

1113957rlowast+1 [33000 33000] 1113957r

lowast+2 [6100 6100]

1113957rlowast+3 [2537300 2537300] 1113957r

lowast+4 [0875 0875] V1 05719

V2 04772

V3 05564

V1 gtV3 gtV21113957rlowastminus1 [37900 37900] 1113957r

lowastminus2 [7100 7100]

1113957rlowastminus3 [2984900 2984900] 1113957r

lowastminus4 [075 075]

e proposed method 1113957rlowast1 [43367 45834] 1113957r

lowast2 [6750 7148]

1113957rlowast3 [2798379 2999621] 1113957r

lowast4 [069 081]

V1 03994

V2 04211

V3 03816

V2 gtV1 gtV3

Mathematical Problems in Engineering 11

accordingly e decision results for different conditions ofreference points are presented in Table 10

As listed in Table 10 when the interval adjustmentcoefficient of the reference points increases gradually theinterval range of the reference points for different attributesis also constantly enlarged so the alternative ranking resultsare not the same As observed in Table 10 if θ isin [0 05] thesorting result is V2 gtV3 gtV1 Moreover if θ isin [075 1] thesorting result is V2 gtV1 gtV3 In addition the alternativeranking results in Table 10 with the continuous expansion ofthe interval for reference points indicate that the compre-hensive prospect value of each alternative also constantlydecreases In addition the differentiation degree of thecomprehensive prospect value between alternatives alsogradually decreases erefore to ensure the effectiveness ofsetting attribute reference points we must consider theinfluence of the interval range of the reference points for thedecision results to reduce the difficulty of decision-makingand improve the accuracy of decision results

6 Conclusion

is paper proposed a new decision-making approach for amixed multiattribute decision-making problem with un-known attribute weights e advantages of this approachare summarized as follows First the reference point for eachattribute is set based on the distribution and comprehensiveambiguity of public expectations making the attributereference points better reflect the publicrsquos expected groupopinions and expectation uncertainty e effectiveness ofthe decision result is guaranteed Second in solving theattribute weights the attribute importance given by thepublic is used to determine the value range of the attributeweights so that the weighting results are in accordance withpublic opinions making the results of the attribute weightmore acceptable en the exact values of the attributeweights are determined to maximize the attribute infor-mation deviation of all the alternatives improving thediscrimination of alternatives ird each alternative is

evaluated based on the prospect theory to satisfy the cal-culation needs of the prospect valuee different expressionforms of the attribute information and reference point areunified and normalization is performed e operationabove can eliminate the influence of the expression form anddimension on decision-making making the decision op-erations smoother

e proposed approach also has certain limitationsFirst in the actual decision-making process the decision-making scenarios in some decision-making problems arenot static Changes in the decision-making scenarios easilycause decision risk which negatively affects the decisionactivity Our proposed approach does not consider dy-namic decision scenarios so the proposed approach mustbe further expanded and improved making it suitable forpublic-participation decision-making problems underchanging scenarios Second according to the settingprinciple of reference points the interval range of the at-tribute reference points affects the decision-making resultse interval range of the attribute reference points is largeand the discrimination of the evaluation results for dif-ferent alternatives is less obvious Because the intervalrange of attribute reference points is set by the compre-hensive ambiguity of public expectations if the dispersiondegree of public expectations is overly high (ie thecomprehensive ambiguity of public expectations is overlyhigh) the interval range of attribute reference points is alsoset over wide corresponding values is outcome may leadto a lack of differentiation in the decision results ereforeto ensure the effectiveness of the decision-making resultsreasonable interval ranges must be set for attribute refer-ence points according to the characteristics of public ex-pectation information which is also worth an in-depthstudy

Data Availability

e data used to support the findings of this study are in-cluded within the manuscript

Table 10 Decision results for different conditions of reference points

θ Reference points Comprehensive prospect value Alternatives ranking

0 1113957rlowast1 [44600 44600] 1113957r

lowast2 [6949 6949]

1113957rlowast3 [2899000 2899000] 1113957r

lowast4 [075 075]

V1 04365

V2 06769

V3 04699V2 gtV3 gtV1

025 1113957rlowast1 [44292 44909] 1113957r

lowast2 [6900 7000]

1113957rlowast3 [2873845 2924155] 1113957r

lowast4 [074 077]

V1 04138

V2 06068

V3 04442V2 gtV3 gtV1

050 1113957rlowast1 [43983 45217] 1113957r

lowast2 [6851 7050]

1113957rlowast3 [2848690 2949311] 1113957r

lowast4 [072 078]

V1 04104

V2 05425

V3 04231V2 gtV3 gtV1

075 1113957rlowast1 [43675 45526] 1113957r

lowast2 [6801 7099]

1113957rlowast3 [2823534 2974466] 1113957r

lowast4 [071 080]

V1 04141

V2 04827

V3 04066V2 gtV1 gtV3

1 1113957rlowast1 [43367 45834] 1113957r

lowast2 [6750 7148]

1113957rlowast3 [2798379 2999621] 1113957r

lowast4 [069 081]

V1 03994

V2 04211

V3 03816V2 gtV1 gtV3

12 Mathematical Problems in Engineering

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is research was funded by the National Natural ScienceFoundation of China (no 71902058) Natural ScienceFoundation of Hunan Province (nos 2018JJ3617 and2021JJ41088) Social Science Foundation of Hunan Province(no 19YBQ113) Scientific Research Foundation of HunanEducation Department (nos 18B484 19A06 and 20B155)and Social Science Achievements Appraisal CommitteeFoundation of Hunan Provincial Department (noXSP21YBC218)

References

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[2] W Chen F Tu and P Zheng ldquoA transnational networkedpublic sphere of air pollution analysis of a Twitter network ofPM25 from the risk society perspectiverdquo InformationCommunication amp Society vol 20 no 7 pp 1005ndash1023 2017

[3] X Yao J He and C Bao ldquoPublic participation modes inChinarsquos environmental impact assessment process an ana-lytical framework based on participation extent and conflictlevelrdquo Environmental Impact Assessment Review vol 84p 106400 2020

[4] X Zhang J G Xu and Y Ju ldquoPublic participation in NIMBYrisk mitigation A discourse zoning approach in the Chinesecontextrdquo Land Use Policy vol 77 pp 559ndash575 2018

[5] T Webler and S Tuler ldquoFour decades of public participationin risk decision makingrdquo Risk Analysis vol 41 no 3pp 503ndash518 2021

[6] J P Voszlig and N Amelung ldquoInnovating public participationmethods Technoscientization and reflexive engagementrdquoSocial Studies of Science vol 46 no 5 pp 749ndash772 2016

[7] A M Rıos B Benito and F Bastida ldquoFactors explainingpublic participation in the central government budget pro-cessrdquo Australian Journal of Public Administration vol 76no 1 pp 48ndash64 2017

[8] Y Zhou L Hou Y Yang H-Y Chong and S Moon ldquoAcomparative review and framework development on publicparticipation for decision-making in Chinese public projectsrdquoEnvironmental Impact Assessment Review vol 75 pp 79ndash872019

[9] T H You J Zhang and Z P Fan ldquoMulti-attribute onlinereview decision making method based on sentiment analysisand evidence theoryrdquo Journal of Systems amp Managementvol 28 no 3 pp 536ndash544 2019 in Chinese

[10] R L Charney T Rebmann P Dalawari and A EndrizalldquoPublic expectations of hospitals to provide resources andservices to the uninjured during disasters A qualitativestudyrdquo Health Security vol 14 no 6 pp 389ndash396 2016

[11] A Yildiz E Ayyildiz A Taskin Gumus and C Ozkan ldquoAframework to prioritize the public expectations from watertreatment plants based on trapezoidal type-2 fuzzy ahpmethodrdquo Environmental Management vol 67 no 3pp 439ndash448 2020

[12] Z Zhang Y Gao and Z L Li ldquoConsensus reaching for socialnetwork group decision making by considering leadershipand bounded confidencerdquo Knowledge-Based Systems vol 204pp 1ndash12 2020

[13] J J Zhu Z Z Ma H H Wang and Y Chen ldquoRisk decision-making method using interval numbers and its applicationbased on the prospect value with multiple reference pointsrdquoInformation Sciences vol 12 no 3 pp 385-386 2017

[14] X Li and X Chen ldquoValue determination method based onmultiple reference points under a trapezoidal intuitionisticfuzzy environmentrdquo Applied Soft Computing vol 63pp 39ndash49 2018

[15] J Gao Z Xu and H Liao ldquoA dynamic reference pointmethod for emergency response under hesitant probabilisticfuzzy environmentrdquo International Journal of Fuzzy Systemsvol 19 no 5 pp 1261ndash1278 2017

[16] C H Li W Li M J Li et al ldquoTarget-oriented model andapproach for attribute value evaluation with multiple refer-ence pointsrdquo Chinese Journal of Management Science vol 25no 7 pp 163ndash175 2017 in Chinese

[17] E Mastrocinque F J Ramırez A Honrubia-Escribano andD T Pham ldquoAn AHP-based multi-criteria model for sus-tainable supply chain development in the renewable energysectorrdquo Expert Systems with Applications vol 150 pp 1ndash172020

[18] L Chen Z Li and X Deng ldquoEmergency alternative evalu-ation under group decision makers a new method based onentropy weight and dematelrdquo International Journal of SystemsScience vol 51 no 3 pp 570ndash583 2020

[19] X KWang Y TWang J QWang P F Cheng and L Li ldquoATODIM-PROMETHEE II based multi-criteria group decisionmaking method for risk evaluation of water resource carryingcapacity under probabilistic linguistic Z-number circum-stancesrdquo Mathematics vol 8 no 7 p 1190 2020

[20] Y Liu Y Dong H Liang F Chiclana and E Herrera-Viedma ldquoMultiple attribute strategic weight manipulationwith minimum cost in a group decision making context withinterval attribute weights informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 49 no 10pp 1981ndash1992 2018

[21] F Meng C Tan and X Chen ldquoAn approach to Atanassovrsquosinterval-valued intuitionistic fuzzy multi-attribute decisionmaking based on prospect theoryrdquo International Journal ofComputational Intelligence Systems vol 8 no 3 pp 591ndash6052015

[22] N Zarbakhshnia Y Wu K Govindan and H Soleimani ldquoAnovel hybrid multiple attribute decision-making approach foroutsourcing sustainable reverse logisticsrdquo Journal of CleanerProduction vol 242 Article ID 118461 2020

[23] J J H Liou Y C Chuang E K Zavadskas and G H TzengldquoData-driven hybrid multiple attribute decision-makingmodel for green supplier evaluation and performance im-provementrdquo Journal of Cleaner Production vol 241 ArticleID 118321 2019

[24] Y H Pan and X L Geng ldquoHybrid multiple attribute decisionmaking approach based on Mo-RVIKORrdquo Chinese Journal ofManagement Science vol 27 no 12 pp 143ndash151 2019

[25] M Akram and D Shumaiza ldquoMulti-criteria decision makingbased on q-rung orthopair fuzzy promethee approachrdquo Ira-nian Journal of Fuzzy Systems vol 18 no 5 pp 107ndash127 2021

[26] C Jana G Muhiuddin and M Pal ldquoMulti-criteria decisionmaking approach based on SVTrN Dombi aggregationfunctionsrdquo Artificial Intelligence Review vol 54 no 5pp 3685ndash3723 2021

Mathematical Problems in Engineering 13

[27] C Jana G Muhiuddin and M Pal ldquoMultiple-attribute de-cision making problems based on SVTNH methodsrdquo Journalof Ambient Intelligence and Humanized Computing vol 11no 9 pp 3717ndash3733 2020

[28] C Jana GMuhiuddin andM Pal ldquoSomeDombi aggregationof Q -rung orthopair fuzzy numbers in multiple-attributedecision makingrdquo International Journal of Intelligent Systemsvol 34 no 12 pp 3220ndash3240 2019

[29] G Shahzadi G Muhiuddin M Arif Butt and A AshrafldquoHamacher interactive hybrid weighted averaging operatorsunder fermatean fuzzy numbersrdquo Journal of Mathematicsvol 2021 no 10 17 pages Article ID 5556017 2021

[30] K Zhang J Zhan and X Wang ldquoTOPSIS-WAA methodbased on a covering-based fuzzy rough set An application torating problemrdquo Information Sciences vol 539 pp 397ndash4212020

[31] S F Huang ldquoUsing Linguistic VIKOR and fuzzy cognitivemaps to select virtual reality games development projectrdquoMathematics vol 9 no 11 p 1253 2021

[32] M Akram A Luqman and C Kahraman ldquoHesitant py-thagorean fuzzy ELECTRE-II method for multi-criteria de-cision-making problemsrdquo Applied Soft Computing vol 108Article ID 107479 2021

[33] C Erdin and G Ozkaya ldquoTurkeyrsquos 2023 energy strategies andinvestment opportunities for renewable energy sources Siteselection based on ELECTRErdquo Sustainability vol 11 no 7Article ID 2136 2019

[34] T Wang H Li X Zhou D Liu and B Huang ldquoree-waydecision based on third-generation prospect theory with Z-numbersrdquo Information Sciences vol 569 pp 13ndash38 2021

[35] Z J Du S M Yu H Y Luo and X D Lin ldquoConsensusconvergence in large-group social network environmentcoordination between trust relationship and opinion simi-larityrdquo Knowledge-Based Systems vol 217 Article ID 1068282021

[36] J Wu F Chiclana H Fujita and E Herrera-Viedma ldquoAvisual interaction consensus model for social network groupdecision making with trust propagationrdquo Knowledge-BasedSystems vol 122 pp 39ndash50 2017

[37] L X Chen and N F Luo ldquoPythagorean fuzzy multi-criteriadecision-making based on prospect theoryrdquo Systems Engi-neering Keory amp Practice vol 40 no 3 pp 726ndash735 2020 inChinese

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[39] R E Moore Methods and Applications of Interval AnalysisPrentice-Hall Hoboken NY USA 1979

[40] R R Ren W W Li M Zhao and X Li ldquoA large groupdecision making method based on public evaluationrdquoManagement Review vol 30 no 10 pp 238ndash247 2018

[41] L Wang Y M Wang and L Martınez ldquoA group decisionmethod based on prospect theory for emergency situationsrdquoInformation Sciences vol 418-419 pp 119ndash135 2017

[42] Y Wu C Xu and T Zhang ldquoEvaluation of renewable powersources using a fuzzy MCDM based on cumulative prospecttheory a case in Chinardquo Energy vol 147 pp 1227ndash1239 2018

[43] M M Salih B B Zaidan A A Zaidan and M A AhmedldquoSurvey on fuzzy TOPSIS state-of-the-art between 2007 and2017rdquo Computers amp Operations Research vol 104 pp 207ndash227 2019

[44] H Xiao Y Zhang X Liu H Yin P Liu and D C Liu ldquoArapid ultrasound vascular disease screening method usingPauTa Criterionrdquo Journal of Physics Conference Seriesvol 1769 Article ID 012009 2021

14 Mathematical Problems in Engineering

expectations We determine the distribution and ambiguityof public expectation according to the expectation infor-mation given by the public and on this basis the referencepoint for each attribute is set

Second we propose an attribute weighting methodconsidering public opinions As we know the value of at-tribute weights has a direct effect on the outcomes of de-cision-making Existing attribute weighting methods rarelyconsider public opinion during the decision-making processin major public affairs To ensure that attribute weightingresult is acceptable to the public we propose an attributeweighting method considering public opinions In this at-tribute weighting method we determine the value range ofattribute weights according to public opinion and establishthe attribute weighting optimization model to determine thefinal attribute weights e result of the attribute weightsobtained using this method reflects the public opinion andguarantees the validity of weighting

ird we introduce the decision-making method of theprospect theory to rank the alternatives in public-partici-pation scenarios e prospect theory considers the un-certainty of events and the decision risk and explains thestructural effect preference nonlinearity resource depen-dence risk pursuit and loss avoidance reflected in peoplersquoschoices [37]erefore the prospect theory is suitable for thedecision-making needs of major public affairs However theexisting research on the prospect theory is not involved inpublic-participation scenarios us the existing decision-making method of prospect theory must be improvedaccording to the characteristics of public participation inmajor public affairs to make it suitable for this type ofdecision-making problem Based on the above analysis weintroduce an improved decision-making method of theprospect theory which requires taking the expectation ofpublic groups as the reference point to guarantee that thedecision-making results better reflect public opinion In thedecision-making process the expression forms of attributeinformation and public information vary to ensure theimplementation of decision-making Different expressionforms of attribute information and public opinions must beunified into an interval before the normalized operation

e rest of the paper is organized as follows In Section 2the preliminaries summarize the knowledge that forms thebasis of the paper and in Section 3 themethod and principleare elucidated explaining the proposed method Section 4presents the case analysis to verify the rationality and validityof the approach In Section 5 a comparison of methods andsensitivity analysis are discussed Finally Section 6 providesthe conclusions

2 Preliminaries

Definition 1 (see [38]) Given a linguistic set S s01113864

s1 sT s0 and sT are the lower and upper limits of thelinguistic variables respectivelye conversion relationshipbetween the linguistic variables st (t 0 1 2 T) andinterval numbers at [aL

t aUt ] is as follows where

0le aLt le aU

t le 1

aL0 0

aUt a

Lt +

1T

0le tleT

aLt a

Utminus 1 1le tleT

aUT 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

Definition 2 Given the uncertain linguistic variables [st1

st2] st1

st2isin S where S s0 s1 sT1113864 1113865

(t1 t2 0 1 2 T 0le t1 le t2 leT) e conversion rela-tionship between the uncertain linguistic variables [st1

st2] and interval number at [aL

t aUt ] (0le aL

t le aUt le

1) is as follows

aLt

t1

T 0le t1 leT

aUt

t2

T 0le t2 leT

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(2)

Definition 3 (see [39]) Suppose that the interval numbera [aL aU] is a nonnegative interval number (ie0le aL le aU) then the midpoint of a is defined bya (aL + aU)2 and the interval number radius of a isdefined by ca (aU minus aL)2

3 Method and Principle

31 Problem Description e decision-making problems inthis study satisfy the following basic assumptions

(1) e set of alternatives is certain(2) e set of attributes to describe the alternatives is

certain(3) e expression form of each attribute is known(4) e public individuals provide expectations based on

their psychological perceptions and the form ofexpression of public expectations is consistent withthe form of expression of the attributes

(5) e public provides the attribute evaluation value ofimportance in the form of linguistic or uncertainlinguistic variables

Suppose that the set of decision-making alternatives for amajor public affair is Z z1 z2 zM1113864 1113865 with attribute setG g1 g2 gN1113864 1113865 and attribute weightω (ω1ω2 ωN)T 1113936

Nj1 ωj 1 e value of attribute j

in alternative i is yij i 1 2 M j 1 2 N More-

over yij can be expressed in the form of crisp numbers

interval numbers linguistic variables or uncertain linguisticvariables e public provides expectation values andevaluates the importance values of the attributes as per theirpreferences Suppose that the number of individuals par-ticipating in the evaluation of the expected value of attribute

Mathematical Problems in Engineering 3

j is Hj e expected value given by individual k on attributej is rk

j where k 1 2 Hj which can be in crispnumbers interval numbers linguistic variables or uncertainlinguistic variables Suppose that the number of public in-dividuals participating in evaluating the importance of at-tribute j is ζj e evaluation value of importance providedby public individual k on attribute j is bk

j wherek 1 2 ζj which can be in linguistic or uncertainlinguistic variables depending on the publicrsquos actual need forattribute expression

e study problem is determining the reference point foreach attribute and the attribute weights according to theattribute values for each alternative the public expectedvalues for each attribute and the public-evaluated impor-tance value for each attribute In addition this study assesseshow to choose a practical decision-making approach to rankall alternatives to determine the optimal alternative

32 Determination of Attribute Reference Points Based onPublic Opinions e expression form of public expectedvalues is primarily related to two factors the attribute ex-pression form of the alternative and the accuracy of indi-vidual expected values If the attribute value of thealternative is expressed in real numbers (crisp numbersinterval numbers etc) the public expected value of theattribute is also expressed in real numbers (crisp numbersinterval numbers etc) In contrast if the attribute value ofthe alternative is expressed in linguistic or uncertain lin-guistic variables the public expected value of the attribute isalso expressed in linguistic or uncertain linguistic variables

e other factor is the accuracy of the individual ex-pected values When the public provides their expectedvalues of an attribute some of the public can express theirexpected values accurately Consequently this portion of thepublic chooses to provide their expected value of the at-tribute in crisp numbers or linguistic variables In contrastthe other portion of the public is affected by internal orexternal factors and cannot accurately express their expectedvalues for the attributes is portion of the public usuallyexpresses their expected values in interval numbers or un-certain linguistic variables

According to the above analysis the expected values ofattributes given by the public also have various forms ofexpression and thus are expressed in different forms eexpression forms must be normalized to determine thespecific distribution and ambiguity of the public expectedvalues and obtain the reference point for each attribute ereference points for attributes can be determined as follows

(1) e initial public expected values of the attribute areprocessed which is achieved as follows

Step 1 e public expected values for the attributefrom linguistic or uncertain linguistic vari-ables are converted into an interval valueusing Definitions 1 and 2

Step 2 e converted public expected values areexpressed in crisp or interval numbersAccording to Definition 3 if the public

individualrsquos expected value rkj is an interval

number described as rkj [rkL

j rkUj ] its

ambiguity is ckj (rkU

j minus rkLj )2 If rk

j is a crispnumber its ambiguity is ck

j 0 e com-prehensive expected ambiguity is defined asthe average expected ambiguity of all publicindividuals for attribute j (iecj (1Hj) 1113936

Hj

k1 ckj)

Step 3 Definition 3 converts the interval form of thepublic individual expected values rk

j intocrisp numbers denoted by rk

j rk

j (rkLj + rkU

j )2 e crisp form of publicexpected values rk

j retains the original formwhich does not need to be converted Afterthe conversion operation the expectationgiven by individual k on attribute j is a crispvalue defined as rk

j

(2) Based on the distribution of rkj the probability

distribution of the expected values on different at-tributes can be determined According to a previousstudy [40] large-scale public opinions usually followa normal distribution

(3) e attribute reference points are determined

Step 1 Under the normal distribution situation forrk

j the mean of public expectation distri-bution over rk

j is defined as μ(rkj ) which can

be determined based on the distribution ofrk

j e distribution variance of the publicexpected value on attribute j is defined asσ(rk

j )Step 2 e attribute reference point 1113957rlowastj is confirmed

which is expressed in the form of intervalnumbers1113957rlowastj [1113957rlowastLj 1113957rlowastUj ] [μ(rk

j ) minus cj μ(rkj) + cj] e

attribute reference point 1113957rlowastj can be deter-mined using the mean of public expectationdistribution μ(rk

j) and the comprehensiveambiguity of the public expected values cj

33 Calculation of the Alternative Prospect Value

331 Normalization of the Attribute Values and ReferencePoints According to Section 32 the finalized form of theattribute reference point is an interval number whereas theattribute value of the alternative can be expressed as a crispnumber interval number linguistic variable or uncertainlinguistic variable e dimensions of various attributes areinconsistent so the attribute values and reference points ofall alternatives must be normalized

e attribute values are unified in the form of intervalnumbers If the attribute value of the alternative is a crispnumber it is rewritten as an interval number with equalupper and lower limits If the attribute value of the al-ternative is a linguistic or uncertain linguistic variablethen Definitions 1 and 2 can convert it into an intervalnumber e interval form of yi

j is defined asyi

j [yiLj yiU

j ]

4 Mathematical Problems in Engineering

Next the attribute values and attribute reference pointsare normalized Equations (3)ndash(6) are used to normalize theattribute values and reference points in interval numbers toeliminate the dimension influence of the original data e

normalized attribute value of yij is

pij [pLij pU

ij] 0lepLij lepU

ij le 1 and the normalized attributereference point of 1113957rlowastj is qj [qL

j qUj ] 0le qL

j le qUj le 1

Attribute gj is a profit index

pLij

yiLj minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

pUij

yiUj minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(3)

qLj

1113957rlowastLj minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

qUj

1113957rlowastUj minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(4)

In addition attribute gj is a cost index

pLij

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus y

iUj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

pUij

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus y

iLj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

qLj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus 1113957r

lowastUj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

qUj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus 1113957r

lowastLj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(6)

332 Calculation of Prospect Profit and Loss Value overDifferent Attributes e profit Gij and loss Fij of attributevalue pij are calculated according to the relationship be-tween the normalized attribute value pij and the nor-malized reference point qj e equations to calculate Gij

and Fij are presented in Table 1 e values of v(+)ij and v

(minus )ij

are determined based on the prospect theory as given inequation (7) According to a previous study [41] the co-efficients of α β λ are α β 088 and λ 225

v(+)ij Gij1113872 1113873

α Gij ge 0

v(minus )ij minus λ minus Fij1113872 1113873

β Fij lt 0

⎧⎪⎨

⎪⎩(7)

e prospect profit-loss matrix of the attribute can beconstructed as follows VM [vij]MtimesN where the value ofvij can be obtained using the following equation

vij v(+)ij + v

(minus )ij (8)

34 Determination of Attribute Weights First the publicevaluation information of the attribute importance is pro-cessed Public individuals give their evaluation values of theimportance of different attributes in the form of linguistic oruncertain linguistic variables Some individuals can expresstheir opinions more accurately so they choose to evaluatethem in linguistic variables Others feel a certain degree ofambiguity or uncertainty over the evaluation results of their

Mathematical Problems in Engineering 5

given attribute importance so they choose to express theiropinions using uncertain linguistic variables According tothe expression characteristics of the public Definitions 1 and2 can be used to convert the public evaluation values fromlinguistic or uncertain linguistic variables into intervalnumbers If the number of individuals participating in theimportance evaluation of attribute j is ζj the importanceevaluation value given by individual k over attribute j isεk

j k 1 2 ζj Based on Definitions 1 and 2 εkj can be

converted into the interval type bkj [bkL

j bkUj ]

k 1 2 ζj We take the average value of the importanceevaluation of the public on attribute j as the comprehensiveimportance evaluation value of attribute j (ie bj [bL

j bUj ]

where bLj (1ζj) 1113936

ζj

k1 bkLk bU

j (1ζj) 1113936ζj

k1 bkUk 0le bL

j lebU

j le 1)Second the value range of the attribute weights ωj is

determined e value range of the attribute weights ωj isassumed to be ωj isin [ωL

j ωUj ] Based on the public com-

prehensive evaluation value of the importance over attributej the upper and lower limits of ωj are determined

ωLj

bLj

bLj + 1113936

Nminus 1e1enejb

Ue

ωUj

bUj

bUj + 1113936

Nminus 1e1enejb

Le

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)

Theorem 1 For ωj isin [ωLj ωU

j ] ωj must exist that meets theconstraints of 0leωj le 1 and 1113936

Nj1 ωj 1

Proof Because 0le bLj le (bL

j + 1113936Nminus 1e1enejb

Ue ) and

0le bUj le (bU

j + 1113936Nminus 1e1enejb

Le ) 0le (bL

j (bLj + 1113936

Nminus 1e1enejb

Ue ))le 1 and

0le (bUj (bU

j + 1113936Nminus 1e1enejb

Le ))le 1 us it can be deduced that

0leωj le 1 As 0le bLj le bU

j le 1 and ωLj (bL

j (bLj + 1113936

Nminus 1e1enej

bUe ))le (bL

j 1113936Nj1 bL

j ) and it can be deduced that (1113936Nj1 ω

Lj

1113936Nj1[bL

j (bLj + 1113936

Nminus 1e1enejb

Ue )])le 1113936

Nj1(bL

j 1113936Nj1 bL

j ) 1 Simi-larly it is deduced that 1113936

Nj1 ω

Uj ge 1 As the value of ωj is

continuous within the range [ωLj ωU

j ] and 0leωLj leωU

j le 1 so1113936

Nj1 ωj 1 must existird we determine the attribute weights An optimi-

zation model is constructed to solve the attribute weights tomaximize the dispersion of attributes on all alternatives

maxψ ωj1113872 1113873 1

M1113944

M

i11113944

N

j1

pLij minus 1113957p

Lj

11138681113868111386811138681113868

11138681113868111386811138681113868 + pUij minus 1113957p

Uj

11138681113868111386811138681113868

11138681113868111386811138681113868

2⎛⎝ ⎞⎠ωj

st

ωLj leωj leω

Uj

1113944

N

j1ωj 1

1113957pj 1113957pLj 1113957p

Uj1113960 1113961

1M

1113944

M

i1p

Lij

1M

1113944

M

i1p

Uij

⎡⎣ ⎤⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)

Table 1 Equations to calculate Gij and Fij

No e relationship between pij and qj e loss Fij e profit Gij

1 pUij lt qL

j 05(pLij + pU

ij) minus qLj 0

2 qUj ltpL

ij 0 05(pLij + pU

ij) minus qUj

3 pLij lt qL

j lepUij lt qU

j 05(pLij minus qL

j ) 0

4 qLj ltpL

ij le qUj ltpU

ij 0 05(pUij minus qU

j )

5 pLij lt qL

j lt qUj ltpU

ij 05(pLij minus qL

j ) 05(pUij minus qU

j )

6 qLj lepL

ij ltpUij le qU

j 0 0

6 Mathematical Problems in Engineering

Theorem 2 Model (10) must have an optimal solution

Proof Under the constraint ωj isin [ωLj ωU

j ] there must be areasonable value of ωj that satisfies 1113936

Nj1 ωj 1 so the

feasible domain of the attribute weight is a nonempty setIn addition 0lepij le 1 and 0lepj le 1 so it is easy to de-duce that 0le ((|pL

ij minus pLj | + |pU

ij minus pUj |)2)le 1 As known

0leωj le 1 1113936Nj1 ωj 1 thus it can be deduced

that 0le 1113936Nj1((|pL

ij minus pLj | + |pU

ij minus pUj |)2)ωj le 1 and 0le

ψ(ωj) (1M) 1113936Mi1 1113936

Nj1 ((|pL

ij minus 1113957pLj |+ |pU

ij minus 1113957pUj |)2)ωj le 1

As ψ(ωj) is a bounded continuous function the constraintcondition of the attribute weight is a bounded closed set soModel (10) must have an optimal solution

To sum up the specific steps of the proposed decision-making approach are as follows

Step 1 e public expected values of attributes withvarious expressions are converted into crisp numbersStep 2e distribution of the public expected values isdetermined on all attributes according to the publicexpected opinion in the crisp number type Next theattribute reference points are obtained based on thedistribution mean of the public expectation andcomprehensive ambiguity of public expected valuesStep 3 e attribute value and reference points arenormalized e attribute prospect value of each al-ternative is calculated based on the prospect theoryStep 4 e value range of the attribute weights isdetermined using equation (9) and Model (10) to de-termine the attribute weightStep 5 e comprehensive prospect values of differentalternatives are obtained using equation (11) to realizethe ranking of alternative alternatives

Vi 1113944N

j1ωjvij (11)

4 Case Analysis

We take a subway construction project as an example toverify the rationality and effectiveness of the method pro-posed in this paper A provincial capital city plans to extendthe No 2 subway line to the west for which three alter-natives can be selected e extension of the subway line willmake public transportation more convenient for residentsalong the line However the subway construction will take along time to be completed occupy a large amount of publicspace and generate substantial dust which interferes withthe daily life of the surrounding residents According to theconstruction requirements and characteristics of the subwayline the organizers of the decision-making activity selectedfour attributes to evaluate the alternatives the averagedistance between the stations and densely populated areasalong the line (g1 units m cost-based index) estimated

construction time (g2 units month cost-based index)enclosed public area for construction (g3 units m

2 cost-based index) and dust and sand treatment effect (g4qualitative index profit-based index) Among these alter-natives g1 is expressed in crisp numbers g2 and g3 areexpressed in interval numbers and g4 is expressed in lin-guistic or uncertain linguistic variables e conversionstandard between the dust and sand treatment effect andlinguistic variables is presented in Table 2 and the attributevalues of different alternatives are listed in Table 3

e primary public group affected by the constructionand operation of the subway is urban residents ereforeduring the decision-making process the opinions of thepublic group directly affected by the subway must be fullyconsidered Various media-driven methods were used topublicize the project to enable the public to understand theactual subway project situation better e public couldexpress their opinions on the subway project using differentmethods such as online platforms telephone and mailquestionnaires e public provides two pieces of evaluationinformation based on their opinions of the attributes theirexpectations and the importance evaluation value econversion standard between the evaluation values of theimportance and linguistic variables is given in Table 2

When the public opinion survey was finished the or-ganizers of the public opinion survey identified and countedthe public individuals who effectively participated estatistical results of the public expectations are listed inTable 4 and the statistical results of the public importanceevaluation values of different attributes are presented inTable 5 Numerous individuals participated in the surveyeffectively Due to the space limitations of the article we onlypresent the partial statistical results of the public opinions inTables 4 and 5

e original public expected values of the attributes inTable 4 were processed First the public expected values ofthe attributes were converted into interval numbers Nextthe comprehensive ambiguity of the expected values of theattributes was calculated which was shown in Table 4 enthe public expected values of the attributes in intervalnumbers were converted into crisp numbers in Definition 3the details of the public expected values for different attri-butes in the form of crisp numbers are shown in Table 6

Finally based on the relevant content in Table 6 thedistribution of public expected values for various attributeswas examined e distribution of public expected values onvarious attributes was examined e fittings of the distri-butions are illustrated in Figures 1ndash4 According to thestatistical distribution results and comprehensive ambiguityof the public expected values the reference point of eachattribute was determined as presented in Table 7

Equations (3)ndash(6) were used to normalize the attributereference points and attribute values of different alternativese normalized attribute reference points and alternativeattribute values are listed in Table 8

Next equations (7) and (8) were used to calculate theprospect values of the attributes and the prospect profit-lossmatrix of the attributes is expressed

Mathematical Problems in Engineering 7

Table 2 Conversion standards between dust and sand treatment (attribute importance) and linguistic variables

Dust and sand treatmenteffect (attribute importance)

Extremely poor (canbe ignored)

Terribly poor(extremely

unimportant)

Very poor (veryunimportant) Poor (unimportant) Fair

(good)

Linguistic variable s0 s1 s2 s3 s4

Dust and sand treatmenteffect (attribute importance) Good (important) Very good (very

important)Extremely good

(extremely important)Perfect (maximum

importance) mdash

Linguistic variable s5 s6 s7 s8 mdash

Table 3 Attribute values of various alternatives

AlternativesAverage distance between the station

and the population gatheringarea along the line (m)

Estimated constructiontime of the

project (months)

Construction enclosedpublic areas (m2)

Dust and sandtreatment

effect (qualitative index)1 330 [64 71] [28732 29849] [S6 S7]2 379 [61 65] [25373 27711] S73 336 [64 68] [28064 29849] [S6 S7]

Table 4 Statistics of public expected values for different attributes

Average distance between thestation and the population

gathering area along the line (m)

Estimated construction time ofthe project (months)

Construction-enclosed publicareas (m2)

Dust and sand treatment effect(qualitative index)

Number of effective participants Number of effective participants Number of effective participants Number of effective participants5472 6070 5518 5491

No Expectedvalues Ambiguity No Expected

value Ambiguity No Expectedvalue Ambiguity No Expected

value Ambiguity

1 [453 499] 23 1 [55 58] 15 1 [3052830789] 1305 1 [S4 S5] 00625

2 [377 377] 0 2 [64 66] 1 2 [2621229284] 1536 2 [S5 S5] 0

5471 [346 352] 3 6069 [67 74] 35 5517 [3004933846] 18985 5490 [S5 S6] 00625

5472 [338 374] 18 6070 [70 78] 4 5518 [2817531285] 1555 5491 [S6 S6] 0

Table 5 Public evaluation results of the importance for different attributes

Average distance betweenthe station and the

population gathering areaalong the line (m)

Estimated constructiontime of the project

(months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

Number of effectiveparticipants

Number of effectiveparticipants

Number of effectiveparticipants

Number of effectiveparticipants

4186 4182 4186 4185No Importance No Importance No Importance No Importance1 [S3 S4] 1 [S1 S2] 1 [S3 S3] 1 [S3 S5]2 [S3 S4] 2 [S1 S2] 2 [S3 S3] 2 [S3 S5]

4185 [S3 S6] 4181 [S2 S2] 4185 [S1 S4] 4184 [S4 S5]4186 [S3 S6] 4182 [S0 S1] 4186 [S1 S4] 4185 [S4 S5]

8 Mathematical Problems in Engineering

Table 6 e public expected values for different attributes in the form of crisp numbers

Average distance betweenthe station and the

population gathering areaalong the line (m)

Estimated construction timeof the project (months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

No Expected value No Expected value No Expected value No Expected value1 476 1 56 1 306585 1 056252 377 2 65 2 27748 2 05000

5471 349 6069 705 5517 319475 5490 068755472 356 6070 74 5518 29730 5491 07500

250 300 350 400 450 500 550 600 6500

20

40

60

80

100

120

140

160

180

Freq

uenc

y

Frequency

Public expectation of attribute 1 (m)

Normal distribution fitting curve

Figure 1 Fitting curve for public expected values of attribute 1

55 60 65 70 75 80 85 900

50

100

150

200

250

300

350

Freq

uenc

y

FrequencyNormal distribution fitting curve

Public expectation of attribute 2 (months)

Figure 2 Fitting curve for public expected values of attribute 2

22 24 26 28 3 32 34 36 380

20

40

60

80

100

120

140

Freq

uenc

y

Frequency

times104

Normal distribution fitting curve

Public expectation of attribute 3 (m2)

Figure 3 Fitting curve for public expected values of attribute 3

05 06 07 08 09 1 11 12Public expectation of attribute 4 (qualitative index)

0

500

1000

1500

2000

2500

Freq

uenc

y

FrequencyNormal distribution fitting curve

Figure 4 Fitting curve for public expected values of attribute 4

Mathematical Problems in Engineering 9

VM

08287 02070 00000 02164

04719 04752 03587 03983

07864 02070 00000 02164

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (12)

e value range of the attribute weights can be deter-mined using equation (9) based on the conversion betweenlinguistic variables and interval numbers ω1 isin [024 055]

ω2 isin [007 026] ω3 isin [010 034]ω4 isin [017 044] Attri-bute weights can be determined using Model (10)ω1 042ω2 007ω3 034ω4 017

Based on the attribute prospect value and attributeweights the comprehensive prospect value of each alter-native can be obtained using equation (11) whereV1 03994 V2 04211 and V3 03816 e ranking ofthree alternatives is V2 gtV1 gtV3 erefore V2 is the op-timal alternative

5 Comparison of Methods andSensitivity Analysis

51ComparisonofMethods To verify the effectiveness of theproposed approach we introduce two existing decision-making methods e first method is the double-referencepoint decision-making method based on the prospect theory[42] e second method is the TOPSIS method [43] In thetwo mentioned methods positive and negative ideal pointsare set as reference points used as the basis of the alternativeevaluation e positive and negative reference points ofattribute j are defined as 1113957rlowast+j and 1113957rlowastminusj respectively Based onthe prospect theory and TOPSIS method the comparisonbetween the attribute value of the alternative and the cor-responding reference points is analyzed and the evaluationvalue of each alternative is obtained In the double-referencepoint decision-making method based on the prospect

theory the maximum and minimum values of public ex-pectation are taken as positive and negative reference points

Public expectations obey the normal distributiontherefore the maximum and minimum values of publicexpectations can be obtained using the three-sigma (3σ)

theorem of normal distribution [44] For attribute j themaximum and minimum values of public expectations areobtained using the 3σ theorem of normal distributiondenoted by μ(rk

j) + 3σ(rkj) and μ(rk

j) minus 3σ(rkj) respectively

According to Definitions 1 and 2 the conversion values oflinguistic variables or uncertain linguistic variables are in therange [0 1] In the conversion crisp valuesrsquo distribution oflinguistic variables or uncertain linguistic variables in orderto ensure the maximum and minimum values we meet therange requirements in this type of distribution we define themaximum and minimum values as follows ifμ(rk

j ) + 3σ(rkj )gt 1 we set maximum value as 1 if

μ(rkj ) minus 3σ(rk

j )lt 0 we set minimum value as 0 For the otherscenarios the maximum and minimum values are set asμ(rk

j ) + 3σ(rkj ) and μ(rk

j) minus 3σ(rkj ) To satisfy the calculation

need of prospect value we set the positive and negativereference points of the attribute as interval numbers withequal upper and lower limits expressed as 1113957rlowast+j [(μ(rk

j) +

3σ(rkj)) (μ(rk

j) + 3σ(rkj))] and 1113957rlowastminusj [(μ(rk

j ) minus 3σ(rkj))

(μ(rkj ) minus 3σ(rk

j ))] In the TOPSISmethod we set the positiveand negative ideal points according to the maximum andminimum values of each attribute in all alternatives Forexample the positive and negative reference points of at-tribute j are respectively expressed as1113957rlowast+j [maxi12MyiU

j maxi12MyiUj ] and

1113957rlowastminusj [mini12MyiLj mini12MyiL

j ] e alternativeranking results of different decision methods are listed inTable 9

According to Table 9 the alternative ranking resultscorresponding to the existing methods proposed are

Table 8 Normalized attribute reference points and attribute values for different alternatives

AlternativeAverage distance between the stationand the population gathering area

along the line (m)

Estimated constructiontime of the project

(months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

1 [10000 10000] [00458 07137] [00318 02734] [03243 10000]2 [06182 06182] [06183 10000] [04943 10000] [10000 10000]3 [09532 09532] [03321 07137] [00318 04179] [03243 10000]Attributereference points [00000 01922] [00000 03798] [00000 04353] [00000 06486]

Table 7 Results of public expectation reference points for different attributes

AttributesAverage distance between the stationand the population gathering area

along the line (m)

Estimatedconstruction time ofthe project (months)

Construction-enclosedpublic areas (m2)

Dust and sandtreatmenteffect

(qualitative index)

Statistical distribution Normal distribution N (4460 74412) Normal distributionN (6949 8482)

Normal distribution N(28990 28992)

Normal distributionN (075 0152)

Comprehensive ambiguityof the expectations 1234 199 100621 006

Attribute reference point [43367 45834] [6750 7148] [2798379 2999621] [069 081]

10 Mathematical Problems in Engineering

inconsistent with the results obtained in this studyemainreason for the inconsistency of decision results is that thereference point setting in each decision method is different

e setting principle of reference points for the twoexisting methods is that reference points are obtainedaccording to the public expectation or the attribute value of thealternative e setting principle for the reference points usedin the two existing methods is relatively simple but it does notconsider the characteristics of public expectations For deci-sion-making activities in major public affairs public expec-tations must be fully considered to ensure the effectiveness ofthe decision-making results erefore public expectationsmust be considered when setting reference points

Based on the above analysis we develop the settingprinciple for reference points by considering public ex-pectations First the information on public expectations isdiscrete and has the characteristics of a normal distributionthus the mean value of public expectations is taken as thestandard setting for the reference point so that the referencepoint can effectively reflect the expectations of the publicgroup Second due to the uncertainty of the expectationevaluation by public individuals the expected value isusually expressed in an interval number which reflects theambiguity of public expectations e comprehensive ex-pected ambiguity of the public is used to represent theuncertain characteristics of public expectations e refer-ence point is set by the mean value of public expectationsand the comprehensive expected ambiguity of the public tobetter reflect the actual situation of public expectation ex-pression e results of the reference points are presented asinterval numbers e value of the reference points reflectsthe expectations of the public group and considers theuncertainty of the public evaluation

If an attribute value in interval form falls within therange of the corresponding reference points in whole or inpart we consider that part of the attribute value that fallswithin the range of the reference points exactly meetspublic expectations which means that the prospect valuefor this part of the attribute is zero erefore when theprospect value is calculated based on the interval reference

point the inclusion or cross relationship between theattribute interval and corresponding reference point in-terval should be considered For example Table 8 indicatesthat the normalized values of attribute 3 for alternatives 1and 3 are p13 [00318 02734] and p33 [00318

04179] respectively Moreover p13 and p33 are not equaland the normalized reference point for attribute 3 isq3 [00000 04353] It is apparent that p13 and p33 areentirely within the interval range of q3 We can affirm thatthe prospect values of attribute 3 for alternatives 1 and 3are both zero (v13 v33 0) In addition we use equations(7) and (8) and Table 8 to calculate the prospect values ofp13 and p33 we can also obtain the same result If wechoose the double-reference point decision-makingmethod to calculate the prospect value of p13 and p33 theresults of the prospect value for p13 and p33 arev13 minus 07329 and v33 minus 06732 which is different fromthe result of the proposed method erefore a differentsetting principle for the reference point leads to differentdecision-making results explaining why the sorting resultsare different in Table 9 erefore if we need to set ref-erence points for decision-making we must set themaccording to the actual situation for the decision problemand decision requirements to guarantee the effectiveness ofthe decision results

52 Sensitivity Analysis of the Reference Point Interval RangeAccording to the content above the comprehensive ambi-guity of public expectations is related to the interval range ofattribute reference points affecting the decision-makingresultse interval adjustment coefficient of reference pointθ (0le θle 1) is introduced in the expression of referencepoints to study the relationship between the value range ofthe reference points and decision results further e ref-erence point of attribute j is defined as1113957rlowastj [1113957rlowastLj 1113957rlowastUj ] [μ(rk

j ) minus θcj μ(rkj) + θcj] e compre-

hensive ambiguity of public expectations for each attribute cj

is known When the values of θ are 0 025 05 075 and 1the corresponding value range of reference points changes

Table 9 Alternative ranking results of different decision methods

Methods Attributereference points Decision results Ranking of

the alternatives

Double-reference point decision-makingmethod based on prospect theory

1113957rlowast+1 [22277 22277] 1113957r

lowast+2 [4405 4405]

1113957rlowast+3 [2029300 2029300] 1113957r

lowast+4 [100 100] V1 minus 01738

V2 minus 00983

V3 minus 01654

V2 gtV3 gtV11113957rlowastminus1 [66923 66923] 1113957r

lowastminus2 [9493 9493]

1113957rlowastminus3 [3768700 3768700] 1113957r

lowastminus4 [000 000]

TOPSIS

1113957rlowast+1 [33000 33000] 1113957r

lowast+2 [6100 6100]

1113957rlowast+3 [2537300 2537300] 1113957r

lowast+4 [0875 0875] V1 05719

V2 04772

V3 05564

V1 gtV3 gtV21113957rlowastminus1 [37900 37900] 1113957r

lowastminus2 [7100 7100]

1113957rlowastminus3 [2984900 2984900] 1113957r

lowastminus4 [075 075]

e proposed method 1113957rlowast1 [43367 45834] 1113957r

lowast2 [6750 7148]

1113957rlowast3 [2798379 2999621] 1113957r

lowast4 [069 081]

V1 03994

V2 04211

V3 03816

V2 gtV1 gtV3

Mathematical Problems in Engineering 11

accordingly e decision results for different conditions ofreference points are presented in Table 10

As listed in Table 10 when the interval adjustmentcoefficient of the reference points increases gradually theinterval range of the reference points for different attributesis also constantly enlarged so the alternative ranking resultsare not the same As observed in Table 10 if θ isin [0 05] thesorting result is V2 gtV3 gtV1 Moreover if θ isin [075 1] thesorting result is V2 gtV1 gtV3 In addition the alternativeranking results in Table 10 with the continuous expansion ofthe interval for reference points indicate that the compre-hensive prospect value of each alternative also constantlydecreases In addition the differentiation degree of thecomprehensive prospect value between alternatives alsogradually decreases erefore to ensure the effectiveness ofsetting attribute reference points we must consider theinfluence of the interval range of the reference points for thedecision results to reduce the difficulty of decision-makingand improve the accuracy of decision results

6 Conclusion

is paper proposed a new decision-making approach for amixed multiattribute decision-making problem with un-known attribute weights e advantages of this approachare summarized as follows First the reference point for eachattribute is set based on the distribution and comprehensiveambiguity of public expectations making the attributereference points better reflect the publicrsquos expected groupopinions and expectation uncertainty e effectiveness ofthe decision result is guaranteed Second in solving theattribute weights the attribute importance given by thepublic is used to determine the value range of the attributeweights so that the weighting results are in accordance withpublic opinions making the results of the attribute weightmore acceptable en the exact values of the attributeweights are determined to maximize the attribute infor-mation deviation of all the alternatives improving thediscrimination of alternatives ird each alternative is

evaluated based on the prospect theory to satisfy the cal-culation needs of the prospect valuee different expressionforms of the attribute information and reference point areunified and normalization is performed e operationabove can eliminate the influence of the expression form anddimension on decision-making making the decision op-erations smoother

e proposed approach also has certain limitationsFirst in the actual decision-making process the decision-making scenarios in some decision-making problems arenot static Changes in the decision-making scenarios easilycause decision risk which negatively affects the decisionactivity Our proposed approach does not consider dy-namic decision scenarios so the proposed approach mustbe further expanded and improved making it suitable forpublic-participation decision-making problems underchanging scenarios Second according to the settingprinciple of reference points the interval range of the at-tribute reference points affects the decision-making resultse interval range of the attribute reference points is largeand the discrimination of the evaluation results for dif-ferent alternatives is less obvious Because the intervalrange of attribute reference points is set by the compre-hensive ambiguity of public expectations if the dispersiondegree of public expectations is overly high (ie thecomprehensive ambiguity of public expectations is overlyhigh) the interval range of attribute reference points is alsoset over wide corresponding values is outcome may leadto a lack of differentiation in the decision results ereforeto ensure the effectiveness of the decision-making resultsreasonable interval ranges must be set for attribute refer-ence points according to the characteristics of public ex-pectation information which is also worth an in-depthstudy

Data Availability

e data used to support the findings of this study are in-cluded within the manuscript

Table 10 Decision results for different conditions of reference points

θ Reference points Comprehensive prospect value Alternatives ranking

0 1113957rlowast1 [44600 44600] 1113957r

lowast2 [6949 6949]

1113957rlowast3 [2899000 2899000] 1113957r

lowast4 [075 075]

V1 04365

V2 06769

V3 04699V2 gtV3 gtV1

025 1113957rlowast1 [44292 44909] 1113957r

lowast2 [6900 7000]

1113957rlowast3 [2873845 2924155] 1113957r

lowast4 [074 077]

V1 04138

V2 06068

V3 04442V2 gtV3 gtV1

050 1113957rlowast1 [43983 45217] 1113957r

lowast2 [6851 7050]

1113957rlowast3 [2848690 2949311] 1113957r

lowast4 [072 078]

V1 04104

V2 05425

V3 04231V2 gtV3 gtV1

075 1113957rlowast1 [43675 45526] 1113957r

lowast2 [6801 7099]

1113957rlowast3 [2823534 2974466] 1113957r

lowast4 [071 080]

V1 04141

V2 04827

V3 04066V2 gtV1 gtV3

1 1113957rlowast1 [43367 45834] 1113957r

lowast2 [6750 7148]

1113957rlowast3 [2798379 2999621] 1113957r

lowast4 [069 081]

V1 03994

V2 04211

V3 03816V2 gtV1 gtV3

12 Mathematical Problems in Engineering

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is research was funded by the National Natural ScienceFoundation of China (no 71902058) Natural ScienceFoundation of Hunan Province (nos 2018JJ3617 and2021JJ41088) Social Science Foundation of Hunan Province(no 19YBQ113) Scientific Research Foundation of HunanEducation Department (nos 18B484 19A06 and 20B155)and Social Science Achievements Appraisal CommitteeFoundation of Hunan Provincial Department (noXSP21YBC218)

References

[1] S Song Z Guo and X Wang ldquoRetracted article the cor-relation between social transformation economic risk andinternet public opinionrdquo Behaviour amp Information Technol-ogy vol 40 no 7 pp 723ndash733 2020

[2] W Chen F Tu and P Zheng ldquoA transnational networkedpublic sphere of air pollution analysis of a Twitter network ofPM25 from the risk society perspectiverdquo InformationCommunication amp Society vol 20 no 7 pp 1005ndash1023 2017

[3] X Yao J He and C Bao ldquoPublic participation modes inChinarsquos environmental impact assessment process an ana-lytical framework based on participation extent and conflictlevelrdquo Environmental Impact Assessment Review vol 84p 106400 2020

[4] X Zhang J G Xu and Y Ju ldquoPublic participation in NIMBYrisk mitigation A discourse zoning approach in the Chinesecontextrdquo Land Use Policy vol 77 pp 559ndash575 2018

[5] T Webler and S Tuler ldquoFour decades of public participationin risk decision makingrdquo Risk Analysis vol 41 no 3pp 503ndash518 2021

[6] J P Voszlig and N Amelung ldquoInnovating public participationmethods Technoscientization and reflexive engagementrdquoSocial Studies of Science vol 46 no 5 pp 749ndash772 2016

[7] A M Rıos B Benito and F Bastida ldquoFactors explainingpublic participation in the central government budget pro-cessrdquo Australian Journal of Public Administration vol 76no 1 pp 48ndash64 2017

[8] Y Zhou L Hou Y Yang H-Y Chong and S Moon ldquoAcomparative review and framework development on publicparticipation for decision-making in Chinese public projectsrdquoEnvironmental Impact Assessment Review vol 75 pp 79ndash872019

[9] T H You J Zhang and Z P Fan ldquoMulti-attribute onlinereview decision making method based on sentiment analysisand evidence theoryrdquo Journal of Systems amp Managementvol 28 no 3 pp 536ndash544 2019 in Chinese

[10] R L Charney T Rebmann P Dalawari and A EndrizalldquoPublic expectations of hospitals to provide resources andservices to the uninjured during disasters A qualitativestudyrdquo Health Security vol 14 no 6 pp 389ndash396 2016

[11] A Yildiz E Ayyildiz A Taskin Gumus and C Ozkan ldquoAframework to prioritize the public expectations from watertreatment plants based on trapezoidal type-2 fuzzy ahpmethodrdquo Environmental Management vol 67 no 3pp 439ndash448 2020

[12] Z Zhang Y Gao and Z L Li ldquoConsensus reaching for socialnetwork group decision making by considering leadershipand bounded confidencerdquo Knowledge-Based Systems vol 204pp 1ndash12 2020

[13] J J Zhu Z Z Ma H H Wang and Y Chen ldquoRisk decision-making method using interval numbers and its applicationbased on the prospect value with multiple reference pointsrdquoInformation Sciences vol 12 no 3 pp 385-386 2017

[14] X Li and X Chen ldquoValue determination method based onmultiple reference points under a trapezoidal intuitionisticfuzzy environmentrdquo Applied Soft Computing vol 63pp 39ndash49 2018

[15] J Gao Z Xu and H Liao ldquoA dynamic reference pointmethod for emergency response under hesitant probabilisticfuzzy environmentrdquo International Journal of Fuzzy Systemsvol 19 no 5 pp 1261ndash1278 2017

[16] C H Li W Li M J Li et al ldquoTarget-oriented model andapproach for attribute value evaluation with multiple refer-ence pointsrdquo Chinese Journal of Management Science vol 25no 7 pp 163ndash175 2017 in Chinese

[17] E Mastrocinque F J Ramırez A Honrubia-Escribano andD T Pham ldquoAn AHP-based multi-criteria model for sus-tainable supply chain development in the renewable energysectorrdquo Expert Systems with Applications vol 150 pp 1ndash172020

[18] L Chen Z Li and X Deng ldquoEmergency alternative evalu-ation under group decision makers a new method based onentropy weight and dematelrdquo International Journal of SystemsScience vol 51 no 3 pp 570ndash583 2020

[19] X KWang Y TWang J QWang P F Cheng and L Li ldquoATODIM-PROMETHEE II based multi-criteria group decisionmaking method for risk evaluation of water resource carryingcapacity under probabilistic linguistic Z-number circum-stancesrdquo Mathematics vol 8 no 7 p 1190 2020

[20] Y Liu Y Dong H Liang F Chiclana and E Herrera-Viedma ldquoMultiple attribute strategic weight manipulationwith minimum cost in a group decision making context withinterval attribute weights informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 49 no 10pp 1981ndash1992 2018

[21] F Meng C Tan and X Chen ldquoAn approach to Atanassovrsquosinterval-valued intuitionistic fuzzy multi-attribute decisionmaking based on prospect theoryrdquo International Journal ofComputational Intelligence Systems vol 8 no 3 pp 591ndash6052015

[22] N Zarbakhshnia Y Wu K Govindan and H Soleimani ldquoAnovel hybrid multiple attribute decision-making approach foroutsourcing sustainable reverse logisticsrdquo Journal of CleanerProduction vol 242 Article ID 118461 2020

[23] J J H Liou Y C Chuang E K Zavadskas and G H TzengldquoData-driven hybrid multiple attribute decision-makingmodel for green supplier evaluation and performance im-provementrdquo Journal of Cleaner Production vol 241 ArticleID 118321 2019

[24] Y H Pan and X L Geng ldquoHybrid multiple attribute decisionmaking approach based on Mo-RVIKORrdquo Chinese Journal ofManagement Science vol 27 no 12 pp 143ndash151 2019

[25] M Akram and D Shumaiza ldquoMulti-criteria decision makingbased on q-rung orthopair fuzzy promethee approachrdquo Ira-nian Journal of Fuzzy Systems vol 18 no 5 pp 107ndash127 2021

[26] C Jana G Muhiuddin and M Pal ldquoMulti-criteria decisionmaking approach based on SVTrN Dombi aggregationfunctionsrdquo Artificial Intelligence Review vol 54 no 5pp 3685ndash3723 2021

Mathematical Problems in Engineering 13

[27] C Jana G Muhiuddin and M Pal ldquoMultiple-attribute de-cision making problems based on SVTNH methodsrdquo Journalof Ambient Intelligence and Humanized Computing vol 11no 9 pp 3717ndash3733 2020

[28] C Jana GMuhiuddin andM Pal ldquoSomeDombi aggregationof Q -rung orthopair fuzzy numbers in multiple-attributedecision makingrdquo International Journal of Intelligent Systemsvol 34 no 12 pp 3220ndash3240 2019

[29] G Shahzadi G Muhiuddin M Arif Butt and A AshrafldquoHamacher interactive hybrid weighted averaging operatorsunder fermatean fuzzy numbersrdquo Journal of Mathematicsvol 2021 no 10 17 pages Article ID 5556017 2021

[30] K Zhang J Zhan and X Wang ldquoTOPSIS-WAA methodbased on a covering-based fuzzy rough set An application torating problemrdquo Information Sciences vol 539 pp 397ndash4212020

[31] S F Huang ldquoUsing Linguistic VIKOR and fuzzy cognitivemaps to select virtual reality games development projectrdquoMathematics vol 9 no 11 p 1253 2021

[32] M Akram A Luqman and C Kahraman ldquoHesitant py-thagorean fuzzy ELECTRE-II method for multi-criteria de-cision-making problemsrdquo Applied Soft Computing vol 108Article ID 107479 2021

[33] C Erdin and G Ozkaya ldquoTurkeyrsquos 2023 energy strategies andinvestment opportunities for renewable energy sources Siteselection based on ELECTRErdquo Sustainability vol 11 no 7Article ID 2136 2019

[34] T Wang H Li X Zhou D Liu and B Huang ldquoree-waydecision based on third-generation prospect theory with Z-numbersrdquo Information Sciences vol 569 pp 13ndash38 2021

[35] Z J Du S M Yu H Y Luo and X D Lin ldquoConsensusconvergence in large-group social network environmentcoordination between trust relationship and opinion simi-larityrdquo Knowledge-Based Systems vol 217 Article ID 1068282021

[36] J Wu F Chiclana H Fujita and E Herrera-Viedma ldquoAvisual interaction consensus model for social network groupdecision making with trust propagationrdquo Knowledge-BasedSystems vol 122 pp 39ndash50 2017

[37] L X Chen and N F Luo ldquoPythagorean fuzzy multi-criteriadecision-making based on prospect theoryrdquo Systems Engi-neering Keory amp Practice vol 40 no 3 pp 726ndash735 2020 inChinese

[38] Z S Xu Linguistic Decision Making Springer-Verlag BerlinGermany 2012

[39] R E Moore Methods and Applications of Interval AnalysisPrentice-Hall Hoboken NY USA 1979

[40] R R Ren W W Li M Zhao and X Li ldquoA large groupdecision making method based on public evaluationrdquoManagement Review vol 30 no 10 pp 238ndash247 2018

[41] L Wang Y M Wang and L Martınez ldquoA group decisionmethod based on prospect theory for emergency situationsrdquoInformation Sciences vol 418-419 pp 119ndash135 2017

[42] Y Wu C Xu and T Zhang ldquoEvaluation of renewable powersources using a fuzzy MCDM based on cumulative prospecttheory a case in Chinardquo Energy vol 147 pp 1227ndash1239 2018

[43] M M Salih B B Zaidan A A Zaidan and M A AhmedldquoSurvey on fuzzy TOPSIS state-of-the-art between 2007 and2017rdquo Computers amp Operations Research vol 104 pp 207ndash227 2019

[44] H Xiao Y Zhang X Liu H Yin P Liu and D C Liu ldquoArapid ultrasound vascular disease screening method usingPauTa Criterionrdquo Journal of Physics Conference Seriesvol 1769 Article ID 012009 2021

14 Mathematical Problems in Engineering

j is Hj e expected value given by individual k on attributej is rk

j where k 1 2 Hj which can be in crispnumbers interval numbers linguistic variables or uncertainlinguistic variables Suppose that the number of public in-dividuals participating in evaluating the importance of at-tribute j is ζj e evaluation value of importance providedby public individual k on attribute j is bk

j wherek 1 2 ζj which can be in linguistic or uncertainlinguistic variables depending on the publicrsquos actual need forattribute expression

e study problem is determining the reference point foreach attribute and the attribute weights according to theattribute values for each alternative the public expectedvalues for each attribute and the public-evaluated impor-tance value for each attribute In addition this study assesseshow to choose a practical decision-making approach to rankall alternatives to determine the optimal alternative

32 Determination of Attribute Reference Points Based onPublic Opinions e expression form of public expectedvalues is primarily related to two factors the attribute ex-pression form of the alternative and the accuracy of indi-vidual expected values If the attribute value of thealternative is expressed in real numbers (crisp numbersinterval numbers etc) the public expected value of theattribute is also expressed in real numbers (crisp numbersinterval numbers etc) In contrast if the attribute value ofthe alternative is expressed in linguistic or uncertain lin-guistic variables the public expected value of the attribute isalso expressed in linguistic or uncertain linguistic variables

e other factor is the accuracy of the individual ex-pected values When the public provides their expectedvalues of an attribute some of the public can express theirexpected values accurately Consequently this portion of thepublic chooses to provide their expected value of the at-tribute in crisp numbers or linguistic variables In contrastthe other portion of the public is affected by internal orexternal factors and cannot accurately express their expectedvalues for the attributes is portion of the public usuallyexpresses their expected values in interval numbers or un-certain linguistic variables

According to the above analysis the expected values ofattributes given by the public also have various forms ofexpression and thus are expressed in different forms eexpression forms must be normalized to determine thespecific distribution and ambiguity of the public expectedvalues and obtain the reference point for each attribute ereference points for attributes can be determined as follows

(1) e initial public expected values of the attribute areprocessed which is achieved as follows

Step 1 e public expected values for the attributefrom linguistic or uncertain linguistic vari-ables are converted into an interval valueusing Definitions 1 and 2

Step 2 e converted public expected values areexpressed in crisp or interval numbersAccording to Definition 3 if the public

individualrsquos expected value rkj is an interval

number described as rkj [rkL

j rkUj ] its

ambiguity is ckj (rkU

j minus rkLj )2 If rk

j is a crispnumber its ambiguity is ck

j 0 e com-prehensive expected ambiguity is defined asthe average expected ambiguity of all publicindividuals for attribute j (iecj (1Hj) 1113936

Hj

k1 ckj)

Step 3 Definition 3 converts the interval form of thepublic individual expected values rk

j intocrisp numbers denoted by rk

j rk

j (rkLj + rkU

j )2 e crisp form of publicexpected values rk

j retains the original formwhich does not need to be converted Afterthe conversion operation the expectationgiven by individual k on attribute j is a crispvalue defined as rk

j

(2) Based on the distribution of rkj the probability

distribution of the expected values on different at-tributes can be determined According to a previousstudy [40] large-scale public opinions usually followa normal distribution

(3) e attribute reference points are determined

Step 1 Under the normal distribution situation forrk

j the mean of public expectation distri-bution over rk

j is defined as μ(rkj ) which can

be determined based on the distribution ofrk

j e distribution variance of the publicexpected value on attribute j is defined asσ(rk

j )Step 2 e attribute reference point 1113957rlowastj is confirmed

which is expressed in the form of intervalnumbers1113957rlowastj [1113957rlowastLj 1113957rlowastUj ] [μ(rk

j ) minus cj μ(rkj) + cj] e

attribute reference point 1113957rlowastj can be deter-mined using the mean of public expectationdistribution μ(rk

j) and the comprehensiveambiguity of the public expected values cj

33 Calculation of the Alternative Prospect Value

331 Normalization of the Attribute Values and ReferencePoints According to Section 32 the finalized form of theattribute reference point is an interval number whereas theattribute value of the alternative can be expressed as a crispnumber interval number linguistic variable or uncertainlinguistic variable e dimensions of various attributes areinconsistent so the attribute values and reference points ofall alternatives must be normalized

e attribute values are unified in the form of intervalnumbers If the attribute value of the alternative is a crispnumber it is rewritten as an interval number with equalupper and lower limits If the attribute value of the al-ternative is a linguistic or uncertain linguistic variablethen Definitions 1 and 2 can convert it into an intervalnumber e interval form of yi

j is defined asyi

j [yiLj yiU

j ]

4 Mathematical Problems in Engineering

Next the attribute values and attribute reference pointsare normalized Equations (3)ndash(6) are used to normalize theattribute values and reference points in interval numbers toeliminate the dimension influence of the original data e

normalized attribute value of yij is

pij [pLij pU

ij] 0lepLij lepU

ij le 1 and the normalized attributereference point of 1113957rlowastj is qj [qL

j qUj ] 0le qL

j le qUj le 1

Attribute gj is a profit index

pLij

yiLj minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

pUij

yiUj minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(3)

qLj

1113957rlowastLj minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

qUj

1113957rlowastUj minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(4)

In addition attribute gj is a cost index

pLij

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus y

iUj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

pUij

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus y

iLj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

qLj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus 1113957r

lowastUj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

qUj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus 1113957r

lowastLj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(6)

332 Calculation of Prospect Profit and Loss Value overDifferent Attributes e profit Gij and loss Fij of attributevalue pij are calculated according to the relationship be-tween the normalized attribute value pij and the nor-malized reference point qj e equations to calculate Gij

and Fij are presented in Table 1 e values of v(+)ij and v

(minus )ij

are determined based on the prospect theory as given inequation (7) According to a previous study [41] the co-efficients of α β λ are α β 088 and λ 225

v(+)ij Gij1113872 1113873

α Gij ge 0

v(minus )ij minus λ minus Fij1113872 1113873

β Fij lt 0

⎧⎪⎨

⎪⎩(7)

e prospect profit-loss matrix of the attribute can beconstructed as follows VM [vij]MtimesN where the value ofvij can be obtained using the following equation

vij v(+)ij + v

(minus )ij (8)

34 Determination of Attribute Weights First the publicevaluation information of the attribute importance is pro-cessed Public individuals give their evaluation values of theimportance of different attributes in the form of linguistic oruncertain linguistic variables Some individuals can expresstheir opinions more accurately so they choose to evaluatethem in linguistic variables Others feel a certain degree ofambiguity or uncertainty over the evaluation results of their

Mathematical Problems in Engineering 5

given attribute importance so they choose to express theiropinions using uncertain linguistic variables According tothe expression characteristics of the public Definitions 1 and2 can be used to convert the public evaluation values fromlinguistic or uncertain linguistic variables into intervalnumbers If the number of individuals participating in theimportance evaluation of attribute j is ζj the importanceevaluation value given by individual k over attribute j isεk

j k 1 2 ζj Based on Definitions 1 and 2 εkj can be

converted into the interval type bkj [bkL

j bkUj ]

k 1 2 ζj We take the average value of the importanceevaluation of the public on attribute j as the comprehensiveimportance evaluation value of attribute j (ie bj [bL

j bUj ]

where bLj (1ζj) 1113936

ζj

k1 bkLk bU

j (1ζj) 1113936ζj

k1 bkUk 0le bL

j lebU

j le 1)Second the value range of the attribute weights ωj is

determined e value range of the attribute weights ωj isassumed to be ωj isin [ωL

j ωUj ] Based on the public com-

prehensive evaluation value of the importance over attributej the upper and lower limits of ωj are determined

ωLj

bLj

bLj + 1113936

Nminus 1e1enejb

Ue

ωUj

bUj

bUj + 1113936

Nminus 1e1enejb

Le

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)

Theorem 1 For ωj isin [ωLj ωU

j ] ωj must exist that meets theconstraints of 0leωj le 1 and 1113936

Nj1 ωj 1

Proof Because 0le bLj le (bL

j + 1113936Nminus 1e1enejb

Ue ) and

0le bUj le (bU

j + 1113936Nminus 1e1enejb

Le ) 0le (bL

j (bLj + 1113936

Nminus 1e1enejb

Ue ))le 1 and

0le (bUj (bU

j + 1113936Nminus 1e1enejb

Le ))le 1 us it can be deduced that

0leωj le 1 As 0le bLj le bU

j le 1 and ωLj (bL

j (bLj + 1113936

Nminus 1e1enej

bUe ))le (bL

j 1113936Nj1 bL

j ) and it can be deduced that (1113936Nj1 ω

Lj

1113936Nj1[bL

j (bLj + 1113936

Nminus 1e1enejb

Ue )])le 1113936

Nj1(bL

j 1113936Nj1 bL

j ) 1 Simi-larly it is deduced that 1113936

Nj1 ω

Uj ge 1 As the value of ωj is

continuous within the range [ωLj ωU

j ] and 0leωLj leωU

j le 1 so1113936

Nj1 ωj 1 must existird we determine the attribute weights An optimi-

zation model is constructed to solve the attribute weights tomaximize the dispersion of attributes on all alternatives

maxψ ωj1113872 1113873 1

M1113944

M

i11113944

N

j1

pLij minus 1113957p

Lj

11138681113868111386811138681113868

11138681113868111386811138681113868 + pUij minus 1113957p

Uj

11138681113868111386811138681113868

11138681113868111386811138681113868

2⎛⎝ ⎞⎠ωj

st

ωLj leωj leω

Uj

1113944

N

j1ωj 1

1113957pj 1113957pLj 1113957p

Uj1113960 1113961

1M

1113944

M

i1p

Lij

1M

1113944

M

i1p

Uij

⎡⎣ ⎤⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)

Table 1 Equations to calculate Gij and Fij

No e relationship between pij and qj e loss Fij e profit Gij

1 pUij lt qL

j 05(pLij + pU

ij) minus qLj 0

2 qUj ltpL

ij 0 05(pLij + pU

ij) minus qUj

3 pLij lt qL

j lepUij lt qU

j 05(pLij minus qL

j ) 0

4 qLj ltpL

ij le qUj ltpU

ij 0 05(pUij minus qU

j )

5 pLij lt qL

j lt qUj ltpU

ij 05(pLij minus qL

j ) 05(pUij minus qU

j )

6 qLj lepL

ij ltpUij le qU

j 0 0

6 Mathematical Problems in Engineering

Theorem 2 Model (10) must have an optimal solution

Proof Under the constraint ωj isin [ωLj ωU

j ] there must be areasonable value of ωj that satisfies 1113936

Nj1 ωj 1 so the

feasible domain of the attribute weight is a nonempty setIn addition 0lepij le 1 and 0lepj le 1 so it is easy to de-duce that 0le ((|pL

ij minus pLj | + |pU

ij minus pUj |)2)le 1 As known

0leωj le 1 1113936Nj1 ωj 1 thus it can be deduced

that 0le 1113936Nj1((|pL

ij minus pLj | + |pU

ij minus pUj |)2)ωj le 1 and 0le

ψ(ωj) (1M) 1113936Mi1 1113936

Nj1 ((|pL

ij minus 1113957pLj |+ |pU

ij minus 1113957pUj |)2)ωj le 1

As ψ(ωj) is a bounded continuous function the constraintcondition of the attribute weight is a bounded closed set soModel (10) must have an optimal solution

To sum up the specific steps of the proposed decision-making approach are as follows

Step 1 e public expected values of attributes withvarious expressions are converted into crisp numbersStep 2e distribution of the public expected values isdetermined on all attributes according to the publicexpected opinion in the crisp number type Next theattribute reference points are obtained based on thedistribution mean of the public expectation andcomprehensive ambiguity of public expected valuesStep 3 e attribute value and reference points arenormalized e attribute prospect value of each al-ternative is calculated based on the prospect theoryStep 4 e value range of the attribute weights isdetermined using equation (9) and Model (10) to de-termine the attribute weightStep 5 e comprehensive prospect values of differentalternatives are obtained using equation (11) to realizethe ranking of alternative alternatives

Vi 1113944N

j1ωjvij (11)

4 Case Analysis

We take a subway construction project as an example toverify the rationality and effectiveness of the method pro-posed in this paper A provincial capital city plans to extendthe No 2 subway line to the west for which three alter-natives can be selected e extension of the subway line willmake public transportation more convenient for residentsalong the line However the subway construction will take along time to be completed occupy a large amount of publicspace and generate substantial dust which interferes withthe daily life of the surrounding residents According to theconstruction requirements and characteristics of the subwayline the organizers of the decision-making activity selectedfour attributes to evaluate the alternatives the averagedistance between the stations and densely populated areasalong the line (g1 units m cost-based index) estimated

construction time (g2 units month cost-based index)enclosed public area for construction (g3 units m

2 cost-based index) and dust and sand treatment effect (g4qualitative index profit-based index) Among these alter-natives g1 is expressed in crisp numbers g2 and g3 areexpressed in interval numbers and g4 is expressed in lin-guistic or uncertain linguistic variables e conversionstandard between the dust and sand treatment effect andlinguistic variables is presented in Table 2 and the attributevalues of different alternatives are listed in Table 3

e primary public group affected by the constructionand operation of the subway is urban residents ereforeduring the decision-making process the opinions of thepublic group directly affected by the subway must be fullyconsidered Various media-driven methods were used topublicize the project to enable the public to understand theactual subway project situation better e public couldexpress their opinions on the subway project using differentmethods such as online platforms telephone and mailquestionnaires e public provides two pieces of evaluationinformation based on their opinions of the attributes theirexpectations and the importance evaluation value econversion standard between the evaluation values of theimportance and linguistic variables is given in Table 2

When the public opinion survey was finished the or-ganizers of the public opinion survey identified and countedthe public individuals who effectively participated estatistical results of the public expectations are listed inTable 4 and the statistical results of the public importanceevaluation values of different attributes are presented inTable 5 Numerous individuals participated in the surveyeffectively Due to the space limitations of the article we onlypresent the partial statistical results of the public opinions inTables 4 and 5

e original public expected values of the attributes inTable 4 were processed First the public expected values ofthe attributes were converted into interval numbers Nextthe comprehensive ambiguity of the expected values of theattributes was calculated which was shown in Table 4 enthe public expected values of the attributes in intervalnumbers were converted into crisp numbers in Definition 3the details of the public expected values for different attri-butes in the form of crisp numbers are shown in Table 6

Finally based on the relevant content in Table 6 thedistribution of public expected values for various attributeswas examined e distribution of public expected values onvarious attributes was examined e fittings of the distri-butions are illustrated in Figures 1ndash4 According to thestatistical distribution results and comprehensive ambiguityof the public expected values the reference point of eachattribute was determined as presented in Table 7

Equations (3)ndash(6) were used to normalize the attributereference points and attribute values of different alternativese normalized attribute reference points and alternativeattribute values are listed in Table 8

Next equations (7) and (8) were used to calculate theprospect values of the attributes and the prospect profit-lossmatrix of the attributes is expressed

Mathematical Problems in Engineering 7

Table 2 Conversion standards between dust and sand treatment (attribute importance) and linguistic variables

Dust and sand treatmenteffect (attribute importance)

Extremely poor (canbe ignored)

Terribly poor(extremely

unimportant)

Very poor (veryunimportant) Poor (unimportant) Fair

(good)

Linguistic variable s0 s1 s2 s3 s4

Dust and sand treatmenteffect (attribute importance) Good (important) Very good (very

important)Extremely good

(extremely important)Perfect (maximum

importance) mdash

Linguistic variable s5 s6 s7 s8 mdash

Table 3 Attribute values of various alternatives

AlternativesAverage distance between the station

and the population gatheringarea along the line (m)

Estimated constructiontime of the

project (months)

Construction enclosedpublic areas (m2)

Dust and sandtreatment

effect (qualitative index)1 330 [64 71] [28732 29849] [S6 S7]2 379 [61 65] [25373 27711] S73 336 [64 68] [28064 29849] [S6 S7]

Table 4 Statistics of public expected values for different attributes

Average distance between thestation and the population

gathering area along the line (m)

Estimated construction time ofthe project (months)

Construction-enclosed publicareas (m2)

Dust and sand treatment effect(qualitative index)

Number of effective participants Number of effective participants Number of effective participants Number of effective participants5472 6070 5518 5491

No Expectedvalues Ambiguity No Expected

value Ambiguity No Expectedvalue Ambiguity No Expected

value Ambiguity

1 [453 499] 23 1 [55 58] 15 1 [3052830789] 1305 1 [S4 S5] 00625

2 [377 377] 0 2 [64 66] 1 2 [2621229284] 1536 2 [S5 S5] 0

5471 [346 352] 3 6069 [67 74] 35 5517 [3004933846] 18985 5490 [S5 S6] 00625

5472 [338 374] 18 6070 [70 78] 4 5518 [2817531285] 1555 5491 [S6 S6] 0

Table 5 Public evaluation results of the importance for different attributes

Average distance betweenthe station and the

population gathering areaalong the line (m)

Estimated constructiontime of the project

(months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

Number of effectiveparticipants

Number of effectiveparticipants

Number of effectiveparticipants

Number of effectiveparticipants

4186 4182 4186 4185No Importance No Importance No Importance No Importance1 [S3 S4] 1 [S1 S2] 1 [S3 S3] 1 [S3 S5]2 [S3 S4] 2 [S1 S2] 2 [S3 S3] 2 [S3 S5]

4185 [S3 S6] 4181 [S2 S2] 4185 [S1 S4] 4184 [S4 S5]4186 [S3 S6] 4182 [S0 S1] 4186 [S1 S4] 4185 [S4 S5]

8 Mathematical Problems in Engineering

Table 6 e public expected values for different attributes in the form of crisp numbers

Average distance betweenthe station and the

population gathering areaalong the line (m)

Estimated construction timeof the project (months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

No Expected value No Expected value No Expected value No Expected value1 476 1 56 1 306585 1 056252 377 2 65 2 27748 2 05000

5471 349 6069 705 5517 319475 5490 068755472 356 6070 74 5518 29730 5491 07500

250 300 350 400 450 500 550 600 6500

20

40

60

80

100

120

140

160

180

Freq

uenc

y

Frequency

Public expectation of attribute 1 (m)

Normal distribution fitting curve

Figure 1 Fitting curve for public expected values of attribute 1

55 60 65 70 75 80 85 900

50

100

150

200

250

300

350

Freq

uenc

y

FrequencyNormal distribution fitting curve

Public expectation of attribute 2 (months)

Figure 2 Fitting curve for public expected values of attribute 2

22 24 26 28 3 32 34 36 380

20

40

60

80

100

120

140

Freq

uenc

y

Frequency

times104

Normal distribution fitting curve

Public expectation of attribute 3 (m2)

Figure 3 Fitting curve for public expected values of attribute 3

05 06 07 08 09 1 11 12Public expectation of attribute 4 (qualitative index)

0

500

1000

1500

2000

2500

Freq

uenc

y

FrequencyNormal distribution fitting curve

Figure 4 Fitting curve for public expected values of attribute 4

Mathematical Problems in Engineering 9

VM

08287 02070 00000 02164

04719 04752 03587 03983

07864 02070 00000 02164

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (12)

e value range of the attribute weights can be deter-mined using equation (9) based on the conversion betweenlinguistic variables and interval numbers ω1 isin [024 055]

ω2 isin [007 026] ω3 isin [010 034]ω4 isin [017 044] Attri-bute weights can be determined using Model (10)ω1 042ω2 007ω3 034ω4 017

Based on the attribute prospect value and attributeweights the comprehensive prospect value of each alter-native can be obtained using equation (11) whereV1 03994 V2 04211 and V3 03816 e ranking ofthree alternatives is V2 gtV1 gtV3 erefore V2 is the op-timal alternative

5 Comparison of Methods andSensitivity Analysis

51ComparisonofMethods To verify the effectiveness of theproposed approach we introduce two existing decision-making methods e first method is the double-referencepoint decision-making method based on the prospect theory[42] e second method is the TOPSIS method [43] In thetwo mentioned methods positive and negative ideal pointsare set as reference points used as the basis of the alternativeevaluation e positive and negative reference points ofattribute j are defined as 1113957rlowast+j and 1113957rlowastminusj respectively Based onthe prospect theory and TOPSIS method the comparisonbetween the attribute value of the alternative and the cor-responding reference points is analyzed and the evaluationvalue of each alternative is obtained In the double-referencepoint decision-making method based on the prospect

theory the maximum and minimum values of public ex-pectation are taken as positive and negative reference points

Public expectations obey the normal distributiontherefore the maximum and minimum values of publicexpectations can be obtained using the three-sigma (3σ)

theorem of normal distribution [44] For attribute j themaximum and minimum values of public expectations areobtained using the 3σ theorem of normal distributiondenoted by μ(rk

j) + 3σ(rkj) and μ(rk

j) minus 3σ(rkj) respectively

According to Definitions 1 and 2 the conversion values oflinguistic variables or uncertain linguistic variables are in therange [0 1] In the conversion crisp valuesrsquo distribution oflinguistic variables or uncertain linguistic variables in orderto ensure the maximum and minimum values we meet therange requirements in this type of distribution we define themaximum and minimum values as follows ifμ(rk

j ) + 3σ(rkj )gt 1 we set maximum value as 1 if

μ(rkj ) minus 3σ(rk

j )lt 0 we set minimum value as 0 For the otherscenarios the maximum and minimum values are set asμ(rk

j ) + 3σ(rkj ) and μ(rk

j) minus 3σ(rkj ) To satisfy the calculation

need of prospect value we set the positive and negativereference points of the attribute as interval numbers withequal upper and lower limits expressed as 1113957rlowast+j [(μ(rk

j) +

3σ(rkj)) (μ(rk

j) + 3σ(rkj))] and 1113957rlowastminusj [(μ(rk

j ) minus 3σ(rkj))

(μ(rkj ) minus 3σ(rk

j ))] In the TOPSISmethod we set the positiveand negative ideal points according to the maximum andminimum values of each attribute in all alternatives Forexample the positive and negative reference points of at-tribute j are respectively expressed as1113957rlowast+j [maxi12MyiU

j maxi12MyiUj ] and

1113957rlowastminusj [mini12MyiLj mini12MyiL

j ] e alternativeranking results of different decision methods are listed inTable 9

According to Table 9 the alternative ranking resultscorresponding to the existing methods proposed are

Table 8 Normalized attribute reference points and attribute values for different alternatives

AlternativeAverage distance between the stationand the population gathering area

along the line (m)

Estimated constructiontime of the project

(months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

1 [10000 10000] [00458 07137] [00318 02734] [03243 10000]2 [06182 06182] [06183 10000] [04943 10000] [10000 10000]3 [09532 09532] [03321 07137] [00318 04179] [03243 10000]Attributereference points [00000 01922] [00000 03798] [00000 04353] [00000 06486]

Table 7 Results of public expectation reference points for different attributes

AttributesAverage distance between the stationand the population gathering area

along the line (m)

Estimatedconstruction time ofthe project (months)

Construction-enclosedpublic areas (m2)

Dust and sandtreatmenteffect

(qualitative index)

Statistical distribution Normal distribution N (4460 74412) Normal distributionN (6949 8482)

Normal distribution N(28990 28992)

Normal distributionN (075 0152)

Comprehensive ambiguityof the expectations 1234 199 100621 006

Attribute reference point [43367 45834] [6750 7148] [2798379 2999621] [069 081]

10 Mathematical Problems in Engineering

inconsistent with the results obtained in this studyemainreason for the inconsistency of decision results is that thereference point setting in each decision method is different

e setting principle of reference points for the twoexisting methods is that reference points are obtainedaccording to the public expectation or the attribute value of thealternative e setting principle for the reference points usedin the two existing methods is relatively simple but it does notconsider the characteristics of public expectations For deci-sion-making activities in major public affairs public expec-tations must be fully considered to ensure the effectiveness ofthe decision-making results erefore public expectationsmust be considered when setting reference points

Based on the above analysis we develop the settingprinciple for reference points by considering public ex-pectations First the information on public expectations isdiscrete and has the characteristics of a normal distributionthus the mean value of public expectations is taken as thestandard setting for the reference point so that the referencepoint can effectively reflect the expectations of the publicgroup Second due to the uncertainty of the expectationevaluation by public individuals the expected value isusually expressed in an interval number which reflects theambiguity of public expectations e comprehensive ex-pected ambiguity of the public is used to represent theuncertain characteristics of public expectations e refer-ence point is set by the mean value of public expectationsand the comprehensive expected ambiguity of the public tobetter reflect the actual situation of public expectation ex-pression e results of the reference points are presented asinterval numbers e value of the reference points reflectsthe expectations of the public group and considers theuncertainty of the public evaluation

If an attribute value in interval form falls within therange of the corresponding reference points in whole or inpart we consider that part of the attribute value that fallswithin the range of the reference points exactly meetspublic expectations which means that the prospect valuefor this part of the attribute is zero erefore when theprospect value is calculated based on the interval reference

point the inclusion or cross relationship between theattribute interval and corresponding reference point in-terval should be considered For example Table 8 indicatesthat the normalized values of attribute 3 for alternatives 1and 3 are p13 [00318 02734] and p33 [00318

04179] respectively Moreover p13 and p33 are not equaland the normalized reference point for attribute 3 isq3 [00000 04353] It is apparent that p13 and p33 areentirely within the interval range of q3 We can affirm thatthe prospect values of attribute 3 for alternatives 1 and 3are both zero (v13 v33 0) In addition we use equations(7) and (8) and Table 8 to calculate the prospect values ofp13 and p33 we can also obtain the same result If wechoose the double-reference point decision-makingmethod to calculate the prospect value of p13 and p33 theresults of the prospect value for p13 and p33 arev13 minus 07329 and v33 minus 06732 which is different fromthe result of the proposed method erefore a differentsetting principle for the reference point leads to differentdecision-making results explaining why the sorting resultsare different in Table 9 erefore if we need to set ref-erence points for decision-making we must set themaccording to the actual situation for the decision problemand decision requirements to guarantee the effectiveness ofthe decision results

52 Sensitivity Analysis of the Reference Point Interval RangeAccording to the content above the comprehensive ambi-guity of public expectations is related to the interval range ofattribute reference points affecting the decision-makingresultse interval adjustment coefficient of reference pointθ (0le θle 1) is introduced in the expression of referencepoints to study the relationship between the value range ofthe reference points and decision results further e ref-erence point of attribute j is defined as1113957rlowastj [1113957rlowastLj 1113957rlowastUj ] [μ(rk

j ) minus θcj μ(rkj) + θcj] e compre-

hensive ambiguity of public expectations for each attribute cj

is known When the values of θ are 0 025 05 075 and 1the corresponding value range of reference points changes

Table 9 Alternative ranking results of different decision methods

Methods Attributereference points Decision results Ranking of

the alternatives

Double-reference point decision-makingmethod based on prospect theory

1113957rlowast+1 [22277 22277] 1113957r

lowast+2 [4405 4405]

1113957rlowast+3 [2029300 2029300] 1113957r

lowast+4 [100 100] V1 minus 01738

V2 minus 00983

V3 minus 01654

V2 gtV3 gtV11113957rlowastminus1 [66923 66923] 1113957r

lowastminus2 [9493 9493]

1113957rlowastminus3 [3768700 3768700] 1113957r

lowastminus4 [000 000]

TOPSIS

1113957rlowast+1 [33000 33000] 1113957r

lowast+2 [6100 6100]

1113957rlowast+3 [2537300 2537300] 1113957r

lowast+4 [0875 0875] V1 05719

V2 04772

V3 05564

V1 gtV3 gtV21113957rlowastminus1 [37900 37900] 1113957r

lowastminus2 [7100 7100]

1113957rlowastminus3 [2984900 2984900] 1113957r

lowastminus4 [075 075]

e proposed method 1113957rlowast1 [43367 45834] 1113957r

lowast2 [6750 7148]

1113957rlowast3 [2798379 2999621] 1113957r

lowast4 [069 081]

V1 03994

V2 04211

V3 03816

V2 gtV1 gtV3

Mathematical Problems in Engineering 11

accordingly e decision results for different conditions ofreference points are presented in Table 10

As listed in Table 10 when the interval adjustmentcoefficient of the reference points increases gradually theinterval range of the reference points for different attributesis also constantly enlarged so the alternative ranking resultsare not the same As observed in Table 10 if θ isin [0 05] thesorting result is V2 gtV3 gtV1 Moreover if θ isin [075 1] thesorting result is V2 gtV1 gtV3 In addition the alternativeranking results in Table 10 with the continuous expansion ofthe interval for reference points indicate that the compre-hensive prospect value of each alternative also constantlydecreases In addition the differentiation degree of thecomprehensive prospect value between alternatives alsogradually decreases erefore to ensure the effectiveness ofsetting attribute reference points we must consider theinfluence of the interval range of the reference points for thedecision results to reduce the difficulty of decision-makingand improve the accuracy of decision results

6 Conclusion

is paper proposed a new decision-making approach for amixed multiattribute decision-making problem with un-known attribute weights e advantages of this approachare summarized as follows First the reference point for eachattribute is set based on the distribution and comprehensiveambiguity of public expectations making the attributereference points better reflect the publicrsquos expected groupopinions and expectation uncertainty e effectiveness ofthe decision result is guaranteed Second in solving theattribute weights the attribute importance given by thepublic is used to determine the value range of the attributeweights so that the weighting results are in accordance withpublic opinions making the results of the attribute weightmore acceptable en the exact values of the attributeweights are determined to maximize the attribute infor-mation deviation of all the alternatives improving thediscrimination of alternatives ird each alternative is

evaluated based on the prospect theory to satisfy the cal-culation needs of the prospect valuee different expressionforms of the attribute information and reference point areunified and normalization is performed e operationabove can eliminate the influence of the expression form anddimension on decision-making making the decision op-erations smoother

e proposed approach also has certain limitationsFirst in the actual decision-making process the decision-making scenarios in some decision-making problems arenot static Changes in the decision-making scenarios easilycause decision risk which negatively affects the decisionactivity Our proposed approach does not consider dy-namic decision scenarios so the proposed approach mustbe further expanded and improved making it suitable forpublic-participation decision-making problems underchanging scenarios Second according to the settingprinciple of reference points the interval range of the at-tribute reference points affects the decision-making resultse interval range of the attribute reference points is largeand the discrimination of the evaluation results for dif-ferent alternatives is less obvious Because the intervalrange of attribute reference points is set by the compre-hensive ambiguity of public expectations if the dispersiondegree of public expectations is overly high (ie thecomprehensive ambiguity of public expectations is overlyhigh) the interval range of attribute reference points is alsoset over wide corresponding values is outcome may leadto a lack of differentiation in the decision results ereforeto ensure the effectiveness of the decision-making resultsreasonable interval ranges must be set for attribute refer-ence points according to the characteristics of public ex-pectation information which is also worth an in-depthstudy

Data Availability

e data used to support the findings of this study are in-cluded within the manuscript

Table 10 Decision results for different conditions of reference points

θ Reference points Comprehensive prospect value Alternatives ranking

0 1113957rlowast1 [44600 44600] 1113957r

lowast2 [6949 6949]

1113957rlowast3 [2899000 2899000] 1113957r

lowast4 [075 075]

V1 04365

V2 06769

V3 04699V2 gtV3 gtV1

025 1113957rlowast1 [44292 44909] 1113957r

lowast2 [6900 7000]

1113957rlowast3 [2873845 2924155] 1113957r

lowast4 [074 077]

V1 04138

V2 06068

V3 04442V2 gtV3 gtV1

050 1113957rlowast1 [43983 45217] 1113957r

lowast2 [6851 7050]

1113957rlowast3 [2848690 2949311] 1113957r

lowast4 [072 078]

V1 04104

V2 05425

V3 04231V2 gtV3 gtV1

075 1113957rlowast1 [43675 45526] 1113957r

lowast2 [6801 7099]

1113957rlowast3 [2823534 2974466] 1113957r

lowast4 [071 080]

V1 04141

V2 04827

V3 04066V2 gtV1 gtV3

1 1113957rlowast1 [43367 45834] 1113957r

lowast2 [6750 7148]

1113957rlowast3 [2798379 2999621] 1113957r

lowast4 [069 081]

V1 03994

V2 04211

V3 03816V2 gtV1 gtV3

12 Mathematical Problems in Engineering

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is research was funded by the National Natural ScienceFoundation of China (no 71902058) Natural ScienceFoundation of Hunan Province (nos 2018JJ3617 and2021JJ41088) Social Science Foundation of Hunan Province(no 19YBQ113) Scientific Research Foundation of HunanEducation Department (nos 18B484 19A06 and 20B155)and Social Science Achievements Appraisal CommitteeFoundation of Hunan Provincial Department (noXSP21YBC218)

References

[1] S Song Z Guo and X Wang ldquoRetracted article the cor-relation between social transformation economic risk andinternet public opinionrdquo Behaviour amp Information Technol-ogy vol 40 no 7 pp 723ndash733 2020

[2] W Chen F Tu and P Zheng ldquoA transnational networkedpublic sphere of air pollution analysis of a Twitter network ofPM25 from the risk society perspectiverdquo InformationCommunication amp Society vol 20 no 7 pp 1005ndash1023 2017

[3] X Yao J He and C Bao ldquoPublic participation modes inChinarsquos environmental impact assessment process an ana-lytical framework based on participation extent and conflictlevelrdquo Environmental Impact Assessment Review vol 84p 106400 2020

[4] X Zhang J G Xu and Y Ju ldquoPublic participation in NIMBYrisk mitigation A discourse zoning approach in the Chinesecontextrdquo Land Use Policy vol 77 pp 559ndash575 2018

[5] T Webler and S Tuler ldquoFour decades of public participationin risk decision makingrdquo Risk Analysis vol 41 no 3pp 503ndash518 2021

[6] J P Voszlig and N Amelung ldquoInnovating public participationmethods Technoscientization and reflexive engagementrdquoSocial Studies of Science vol 46 no 5 pp 749ndash772 2016

[7] A M Rıos B Benito and F Bastida ldquoFactors explainingpublic participation in the central government budget pro-cessrdquo Australian Journal of Public Administration vol 76no 1 pp 48ndash64 2017

[8] Y Zhou L Hou Y Yang H-Y Chong and S Moon ldquoAcomparative review and framework development on publicparticipation for decision-making in Chinese public projectsrdquoEnvironmental Impact Assessment Review vol 75 pp 79ndash872019

[9] T H You J Zhang and Z P Fan ldquoMulti-attribute onlinereview decision making method based on sentiment analysisand evidence theoryrdquo Journal of Systems amp Managementvol 28 no 3 pp 536ndash544 2019 in Chinese

[10] R L Charney T Rebmann P Dalawari and A EndrizalldquoPublic expectations of hospitals to provide resources andservices to the uninjured during disasters A qualitativestudyrdquo Health Security vol 14 no 6 pp 389ndash396 2016

[11] A Yildiz E Ayyildiz A Taskin Gumus and C Ozkan ldquoAframework to prioritize the public expectations from watertreatment plants based on trapezoidal type-2 fuzzy ahpmethodrdquo Environmental Management vol 67 no 3pp 439ndash448 2020

[12] Z Zhang Y Gao and Z L Li ldquoConsensus reaching for socialnetwork group decision making by considering leadershipand bounded confidencerdquo Knowledge-Based Systems vol 204pp 1ndash12 2020

[13] J J Zhu Z Z Ma H H Wang and Y Chen ldquoRisk decision-making method using interval numbers and its applicationbased on the prospect value with multiple reference pointsrdquoInformation Sciences vol 12 no 3 pp 385-386 2017

[14] X Li and X Chen ldquoValue determination method based onmultiple reference points under a trapezoidal intuitionisticfuzzy environmentrdquo Applied Soft Computing vol 63pp 39ndash49 2018

[15] J Gao Z Xu and H Liao ldquoA dynamic reference pointmethod for emergency response under hesitant probabilisticfuzzy environmentrdquo International Journal of Fuzzy Systemsvol 19 no 5 pp 1261ndash1278 2017

[16] C H Li W Li M J Li et al ldquoTarget-oriented model andapproach for attribute value evaluation with multiple refer-ence pointsrdquo Chinese Journal of Management Science vol 25no 7 pp 163ndash175 2017 in Chinese

[17] E Mastrocinque F J Ramırez A Honrubia-Escribano andD T Pham ldquoAn AHP-based multi-criteria model for sus-tainable supply chain development in the renewable energysectorrdquo Expert Systems with Applications vol 150 pp 1ndash172020

[18] L Chen Z Li and X Deng ldquoEmergency alternative evalu-ation under group decision makers a new method based onentropy weight and dematelrdquo International Journal of SystemsScience vol 51 no 3 pp 570ndash583 2020

[19] X KWang Y TWang J QWang P F Cheng and L Li ldquoATODIM-PROMETHEE II based multi-criteria group decisionmaking method for risk evaluation of water resource carryingcapacity under probabilistic linguistic Z-number circum-stancesrdquo Mathematics vol 8 no 7 p 1190 2020

[20] Y Liu Y Dong H Liang F Chiclana and E Herrera-Viedma ldquoMultiple attribute strategic weight manipulationwith minimum cost in a group decision making context withinterval attribute weights informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 49 no 10pp 1981ndash1992 2018

[21] F Meng C Tan and X Chen ldquoAn approach to Atanassovrsquosinterval-valued intuitionistic fuzzy multi-attribute decisionmaking based on prospect theoryrdquo International Journal ofComputational Intelligence Systems vol 8 no 3 pp 591ndash6052015

[22] N Zarbakhshnia Y Wu K Govindan and H Soleimani ldquoAnovel hybrid multiple attribute decision-making approach foroutsourcing sustainable reverse logisticsrdquo Journal of CleanerProduction vol 242 Article ID 118461 2020

[23] J J H Liou Y C Chuang E K Zavadskas and G H TzengldquoData-driven hybrid multiple attribute decision-makingmodel for green supplier evaluation and performance im-provementrdquo Journal of Cleaner Production vol 241 ArticleID 118321 2019

[24] Y H Pan and X L Geng ldquoHybrid multiple attribute decisionmaking approach based on Mo-RVIKORrdquo Chinese Journal ofManagement Science vol 27 no 12 pp 143ndash151 2019

[25] M Akram and D Shumaiza ldquoMulti-criteria decision makingbased on q-rung orthopair fuzzy promethee approachrdquo Ira-nian Journal of Fuzzy Systems vol 18 no 5 pp 107ndash127 2021

[26] C Jana G Muhiuddin and M Pal ldquoMulti-criteria decisionmaking approach based on SVTrN Dombi aggregationfunctionsrdquo Artificial Intelligence Review vol 54 no 5pp 3685ndash3723 2021

Mathematical Problems in Engineering 13

[27] C Jana G Muhiuddin and M Pal ldquoMultiple-attribute de-cision making problems based on SVTNH methodsrdquo Journalof Ambient Intelligence and Humanized Computing vol 11no 9 pp 3717ndash3733 2020

[28] C Jana GMuhiuddin andM Pal ldquoSomeDombi aggregationof Q -rung orthopair fuzzy numbers in multiple-attributedecision makingrdquo International Journal of Intelligent Systemsvol 34 no 12 pp 3220ndash3240 2019

[29] G Shahzadi G Muhiuddin M Arif Butt and A AshrafldquoHamacher interactive hybrid weighted averaging operatorsunder fermatean fuzzy numbersrdquo Journal of Mathematicsvol 2021 no 10 17 pages Article ID 5556017 2021

[30] K Zhang J Zhan and X Wang ldquoTOPSIS-WAA methodbased on a covering-based fuzzy rough set An application torating problemrdquo Information Sciences vol 539 pp 397ndash4212020

[31] S F Huang ldquoUsing Linguistic VIKOR and fuzzy cognitivemaps to select virtual reality games development projectrdquoMathematics vol 9 no 11 p 1253 2021

[32] M Akram A Luqman and C Kahraman ldquoHesitant py-thagorean fuzzy ELECTRE-II method for multi-criteria de-cision-making problemsrdquo Applied Soft Computing vol 108Article ID 107479 2021

[33] C Erdin and G Ozkaya ldquoTurkeyrsquos 2023 energy strategies andinvestment opportunities for renewable energy sources Siteselection based on ELECTRErdquo Sustainability vol 11 no 7Article ID 2136 2019

[34] T Wang H Li X Zhou D Liu and B Huang ldquoree-waydecision based on third-generation prospect theory with Z-numbersrdquo Information Sciences vol 569 pp 13ndash38 2021

[35] Z J Du S M Yu H Y Luo and X D Lin ldquoConsensusconvergence in large-group social network environmentcoordination between trust relationship and opinion simi-larityrdquo Knowledge-Based Systems vol 217 Article ID 1068282021

[36] J Wu F Chiclana H Fujita and E Herrera-Viedma ldquoAvisual interaction consensus model for social network groupdecision making with trust propagationrdquo Knowledge-BasedSystems vol 122 pp 39ndash50 2017

[37] L X Chen and N F Luo ldquoPythagorean fuzzy multi-criteriadecision-making based on prospect theoryrdquo Systems Engi-neering Keory amp Practice vol 40 no 3 pp 726ndash735 2020 inChinese

[38] Z S Xu Linguistic Decision Making Springer-Verlag BerlinGermany 2012

[39] R E Moore Methods and Applications of Interval AnalysisPrentice-Hall Hoboken NY USA 1979

[40] R R Ren W W Li M Zhao and X Li ldquoA large groupdecision making method based on public evaluationrdquoManagement Review vol 30 no 10 pp 238ndash247 2018

[41] L Wang Y M Wang and L Martınez ldquoA group decisionmethod based on prospect theory for emergency situationsrdquoInformation Sciences vol 418-419 pp 119ndash135 2017

[42] Y Wu C Xu and T Zhang ldquoEvaluation of renewable powersources using a fuzzy MCDM based on cumulative prospecttheory a case in Chinardquo Energy vol 147 pp 1227ndash1239 2018

[43] M M Salih B B Zaidan A A Zaidan and M A AhmedldquoSurvey on fuzzy TOPSIS state-of-the-art between 2007 and2017rdquo Computers amp Operations Research vol 104 pp 207ndash227 2019

[44] H Xiao Y Zhang X Liu H Yin P Liu and D C Liu ldquoArapid ultrasound vascular disease screening method usingPauTa Criterionrdquo Journal of Physics Conference Seriesvol 1769 Article ID 012009 2021

14 Mathematical Problems in Engineering

Next the attribute values and attribute reference pointsare normalized Equations (3)ndash(6) are used to normalize theattribute values and reference points in interval numbers toeliminate the dimension influence of the original data e

normalized attribute value of yij is

pij [pLij pU

ij] 0lepLij lepU

ij le 1 and the normalized attributereference point of 1113957rlowastj is qj [qL

j qUj ] 0le qL

j le qUj le 1

Attribute gj is a profit index

pLij

yiLj minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

pUij

yiUj minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(3)

qLj

1113957rlowastLj minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

qUj

1113957rlowastUj minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(4)

In addition attribute gj is a cost index

pLij

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus y

iUj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

pUij

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus y

iLj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

qLj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus 1113957r

lowastUj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

qUj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus 1113957r

lowastLj

max max1leileM yiUj1113966 1113967 1113957rlowastUj1113966 1113967 minus min min1leileM y

iLj1113966 1113967 1113957rlowastLj1113966 1113967

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(6)

332 Calculation of Prospect Profit and Loss Value overDifferent Attributes e profit Gij and loss Fij of attributevalue pij are calculated according to the relationship be-tween the normalized attribute value pij and the nor-malized reference point qj e equations to calculate Gij

and Fij are presented in Table 1 e values of v(+)ij and v

(minus )ij

are determined based on the prospect theory as given inequation (7) According to a previous study [41] the co-efficients of α β λ are α β 088 and λ 225

v(+)ij Gij1113872 1113873

α Gij ge 0

v(minus )ij minus λ minus Fij1113872 1113873

β Fij lt 0

⎧⎪⎨

⎪⎩(7)

e prospect profit-loss matrix of the attribute can beconstructed as follows VM [vij]MtimesN where the value ofvij can be obtained using the following equation

vij v(+)ij + v

(minus )ij (8)

34 Determination of Attribute Weights First the publicevaluation information of the attribute importance is pro-cessed Public individuals give their evaluation values of theimportance of different attributes in the form of linguistic oruncertain linguistic variables Some individuals can expresstheir opinions more accurately so they choose to evaluatethem in linguistic variables Others feel a certain degree ofambiguity or uncertainty over the evaluation results of their

Mathematical Problems in Engineering 5

given attribute importance so they choose to express theiropinions using uncertain linguistic variables According tothe expression characteristics of the public Definitions 1 and2 can be used to convert the public evaluation values fromlinguistic or uncertain linguistic variables into intervalnumbers If the number of individuals participating in theimportance evaluation of attribute j is ζj the importanceevaluation value given by individual k over attribute j isεk

j k 1 2 ζj Based on Definitions 1 and 2 εkj can be

converted into the interval type bkj [bkL

j bkUj ]

k 1 2 ζj We take the average value of the importanceevaluation of the public on attribute j as the comprehensiveimportance evaluation value of attribute j (ie bj [bL

j bUj ]

where bLj (1ζj) 1113936

ζj

k1 bkLk bU

j (1ζj) 1113936ζj

k1 bkUk 0le bL

j lebU

j le 1)Second the value range of the attribute weights ωj is

determined e value range of the attribute weights ωj isassumed to be ωj isin [ωL

j ωUj ] Based on the public com-

prehensive evaluation value of the importance over attributej the upper and lower limits of ωj are determined

ωLj

bLj

bLj + 1113936

Nminus 1e1enejb

Ue

ωUj

bUj

bUj + 1113936

Nminus 1e1enejb

Le

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)

Theorem 1 For ωj isin [ωLj ωU

j ] ωj must exist that meets theconstraints of 0leωj le 1 and 1113936

Nj1 ωj 1

Proof Because 0le bLj le (bL

j + 1113936Nminus 1e1enejb

Ue ) and

0le bUj le (bU

j + 1113936Nminus 1e1enejb

Le ) 0le (bL

j (bLj + 1113936

Nminus 1e1enejb

Ue ))le 1 and

0le (bUj (bU

j + 1113936Nminus 1e1enejb

Le ))le 1 us it can be deduced that

0leωj le 1 As 0le bLj le bU

j le 1 and ωLj (bL

j (bLj + 1113936

Nminus 1e1enej

bUe ))le (bL

j 1113936Nj1 bL

j ) and it can be deduced that (1113936Nj1 ω

Lj

1113936Nj1[bL

j (bLj + 1113936

Nminus 1e1enejb

Ue )])le 1113936

Nj1(bL

j 1113936Nj1 bL

j ) 1 Simi-larly it is deduced that 1113936

Nj1 ω

Uj ge 1 As the value of ωj is

continuous within the range [ωLj ωU

j ] and 0leωLj leωU

j le 1 so1113936

Nj1 ωj 1 must existird we determine the attribute weights An optimi-

zation model is constructed to solve the attribute weights tomaximize the dispersion of attributes on all alternatives

maxψ ωj1113872 1113873 1

M1113944

M

i11113944

N

j1

pLij minus 1113957p

Lj

11138681113868111386811138681113868

11138681113868111386811138681113868 + pUij minus 1113957p

Uj

11138681113868111386811138681113868

11138681113868111386811138681113868

2⎛⎝ ⎞⎠ωj

st

ωLj leωj leω

Uj

1113944

N

j1ωj 1

1113957pj 1113957pLj 1113957p

Uj1113960 1113961

1M

1113944

M

i1p

Lij

1M

1113944

M

i1p

Uij

⎡⎣ ⎤⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)

Table 1 Equations to calculate Gij and Fij

No e relationship between pij and qj e loss Fij e profit Gij

1 pUij lt qL

j 05(pLij + pU

ij) minus qLj 0

2 qUj ltpL

ij 0 05(pLij + pU

ij) minus qUj

3 pLij lt qL

j lepUij lt qU

j 05(pLij minus qL

j ) 0

4 qLj ltpL

ij le qUj ltpU

ij 0 05(pUij minus qU

j )

5 pLij lt qL

j lt qUj ltpU

ij 05(pLij minus qL

j ) 05(pUij minus qU

j )

6 qLj lepL

ij ltpUij le qU

j 0 0

6 Mathematical Problems in Engineering

Theorem 2 Model (10) must have an optimal solution

Proof Under the constraint ωj isin [ωLj ωU

j ] there must be areasonable value of ωj that satisfies 1113936

Nj1 ωj 1 so the

feasible domain of the attribute weight is a nonempty setIn addition 0lepij le 1 and 0lepj le 1 so it is easy to de-duce that 0le ((|pL

ij minus pLj | + |pU

ij minus pUj |)2)le 1 As known

0leωj le 1 1113936Nj1 ωj 1 thus it can be deduced

that 0le 1113936Nj1((|pL

ij minus pLj | + |pU

ij minus pUj |)2)ωj le 1 and 0le

ψ(ωj) (1M) 1113936Mi1 1113936

Nj1 ((|pL

ij minus 1113957pLj |+ |pU

ij minus 1113957pUj |)2)ωj le 1

As ψ(ωj) is a bounded continuous function the constraintcondition of the attribute weight is a bounded closed set soModel (10) must have an optimal solution

To sum up the specific steps of the proposed decision-making approach are as follows

Step 1 e public expected values of attributes withvarious expressions are converted into crisp numbersStep 2e distribution of the public expected values isdetermined on all attributes according to the publicexpected opinion in the crisp number type Next theattribute reference points are obtained based on thedistribution mean of the public expectation andcomprehensive ambiguity of public expected valuesStep 3 e attribute value and reference points arenormalized e attribute prospect value of each al-ternative is calculated based on the prospect theoryStep 4 e value range of the attribute weights isdetermined using equation (9) and Model (10) to de-termine the attribute weightStep 5 e comprehensive prospect values of differentalternatives are obtained using equation (11) to realizethe ranking of alternative alternatives

Vi 1113944N

j1ωjvij (11)

4 Case Analysis

We take a subway construction project as an example toverify the rationality and effectiveness of the method pro-posed in this paper A provincial capital city plans to extendthe No 2 subway line to the west for which three alter-natives can be selected e extension of the subway line willmake public transportation more convenient for residentsalong the line However the subway construction will take along time to be completed occupy a large amount of publicspace and generate substantial dust which interferes withthe daily life of the surrounding residents According to theconstruction requirements and characteristics of the subwayline the organizers of the decision-making activity selectedfour attributes to evaluate the alternatives the averagedistance between the stations and densely populated areasalong the line (g1 units m cost-based index) estimated

construction time (g2 units month cost-based index)enclosed public area for construction (g3 units m

2 cost-based index) and dust and sand treatment effect (g4qualitative index profit-based index) Among these alter-natives g1 is expressed in crisp numbers g2 and g3 areexpressed in interval numbers and g4 is expressed in lin-guistic or uncertain linguistic variables e conversionstandard between the dust and sand treatment effect andlinguistic variables is presented in Table 2 and the attributevalues of different alternatives are listed in Table 3

e primary public group affected by the constructionand operation of the subway is urban residents ereforeduring the decision-making process the opinions of thepublic group directly affected by the subway must be fullyconsidered Various media-driven methods were used topublicize the project to enable the public to understand theactual subway project situation better e public couldexpress their opinions on the subway project using differentmethods such as online platforms telephone and mailquestionnaires e public provides two pieces of evaluationinformation based on their opinions of the attributes theirexpectations and the importance evaluation value econversion standard between the evaluation values of theimportance and linguistic variables is given in Table 2

When the public opinion survey was finished the or-ganizers of the public opinion survey identified and countedthe public individuals who effectively participated estatistical results of the public expectations are listed inTable 4 and the statistical results of the public importanceevaluation values of different attributes are presented inTable 5 Numerous individuals participated in the surveyeffectively Due to the space limitations of the article we onlypresent the partial statistical results of the public opinions inTables 4 and 5

e original public expected values of the attributes inTable 4 were processed First the public expected values ofthe attributes were converted into interval numbers Nextthe comprehensive ambiguity of the expected values of theattributes was calculated which was shown in Table 4 enthe public expected values of the attributes in intervalnumbers were converted into crisp numbers in Definition 3the details of the public expected values for different attri-butes in the form of crisp numbers are shown in Table 6

Finally based on the relevant content in Table 6 thedistribution of public expected values for various attributeswas examined e distribution of public expected values onvarious attributes was examined e fittings of the distri-butions are illustrated in Figures 1ndash4 According to thestatistical distribution results and comprehensive ambiguityof the public expected values the reference point of eachattribute was determined as presented in Table 7

Equations (3)ndash(6) were used to normalize the attributereference points and attribute values of different alternativese normalized attribute reference points and alternativeattribute values are listed in Table 8

Next equations (7) and (8) were used to calculate theprospect values of the attributes and the prospect profit-lossmatrix of the attributes is expressed

Mathematical Problems in Engineering 7

Table 2 Conversion standards between dust and sand treatment (attribute importance) and linguistic variables

Dust and sand treatmenteffect (attribute importance)

Extremely poor (canbe ignored)

Terribly poor(extremely

unimportant)

Very poor (veryunimportant) Poor (unimportant) Fair

(good)

Linguistic variable s0 s1 s2 s3 s4

Dust and sand treatmenteffect (attribute importance) Good (important) Very good (very

important)Extremely good

(extremely important)Perfect (maximum

importance) mdash

Linguistic variable s5 s6 s7 s8 mdash

Table 3 Attribute values of various alternatives

AlternativesAverage distance between the station

and the population gatheringarea along the line (m)

Estimated constructiontime of the

project (months)

Construction enclosedpublic areas (m2)

Dust and sandtreatment

effect (qualitative index)1 330 [64 71] [28732 29849] [S6 S7]2 379 [61 65] [25373 27711] S73 336 [64 68] [28064 29849] [S6 S7]

Table 4 Statistics of public expected values for different attributes

Average distance between thestation and the population

gathering area along the line (m)

Estimated construction time ofthe project (months)

Construction-enclosed publicareas (m2)

Dust and sand treatment effect(qualitative index)

Number of effective participants Number of effective participants Number of effective participants Number of effective participants5472 6070 5518 5491

No Expectedvalues Ambiguity No Expected

value Ambiguity No Expectedvalue Ambiguity No Expected

value Ambiguity

1 [453 499] 23 1 [55 58] 15 1 [3052830789] 1305 1 [S4 S5] 00625

2 [377 377] 0 2 [64 66] 1 2 [2621229284] 1536 2 [S5 S5] 0

5471 [346 352] 3 6069 [67 74] 35 5517 [3004933846] 18985 5490 [S5 S6] 00625

5472 [338 374] 18 6070 [70 78] 4 5518 [2817531285] 1555 5491 [S6 S6] 0

Table 5 Public evaluation results of the importance for different attributes

Average distance betweenthe station and the

population gathering areaalong the line (m)

Estimated constructiontime of the project

(months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

Number of effectiveparticipants

Number of effectiveparticipants

Number of effectiveparticipants

Number of effectiveparticipants

4186 4182 4186 4185No Importance No Importance No Importance No Importance1 [S3 S4] 1 [S1 S2] 1 [S3 S3] 1 [S3 S5]2 [S3 S4] 2 [S1 S2] 2 [S3 S3] 2 [S3 S5]

4185 [S3 S6] 4181 [S2 S2] 4185 [S1 S4] 4184 [S4 S5]4186 [S3 S6] 4182 [S0 S1] 4186 [S1 S4] 4185 [S4 S5]

8 Mathematical Problems in Engineering

Table 6 e public expected values for different attributes in the form of crisp numbers

Average distance betweenthe station and the

population gathering areaalong the line (m)

Estimated construction timeof the project (months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

No Expected value No Expected value No Expected value No Expected value1 476 1 56 1 306585 1 056252 377 2 65 2 27748 2 05000

5471 349 6069 705 5517 319475 5490 068755472 356 6070 74 5518 29730 5491 07500

250 300 350 400 450 500 550 600 6500

20

40

60

80

100

120

140

160

180

Freq

uenc

y

Frequency

Public expectation of attribute 1 (m)

Normal distribution fitting curve

Figure 1 Fitting curve for public expected values of attribute 1

55 60 65 70 75 80 85 900

50

100

150

200

250

300

350

Freq

uenc

y

FrequencyNormal distribution fitting curve

Public expectation of attribute 2 (months)

Figure 2 Fitting curve for public expected values of attribute 2

22 24 26 28 3 32 34 36 380

20

40

60

80

100

120

140

Freq

uenc

y

Frequency

times104

Normal distribution fitting curve

Public expectation of attribute 3 (m2)

Figure 3 Fitting curve for public expected values of attribute 3

05 06 07 08 09 1 11 12Public expectation of attribute 4 (qualitative index)

0

500

1000

1500

2000

2500

Freq

uenc

y

FrequencyNormal distribution fitting curve

Figure 4 Fitting curve for public expected values of attribute 4

Mathematical Problems in Engineering 9

VM

08287 02070 00000 02164

04719 04752 03587 03983

07864 02070 00000 02164

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (12)

e value range of the attribute weights can be deter-mined using equation (9) based on the conversion betweenlinguistic variables and interval numbers ω1 isin [024 055]

ω2 isin [007 026] ω3 isin [010 034]ω4 isin [017 044] Attri-bute weights can be determined using Model (10)ω1 042ω2 007ω3 034ω4 017

Based on the attribute prospect value and attributeweights the comprehensive prospect value of each alter-native can be obtained using equation (11) whereV1 03994 V2 04211 and V3 03816 e ranking ofthree alternatives is V2 gtV1 gtV3 erefore V2 is the op-timal alternative

5 Comparison of Methods andSensitivity Analysis

51ComparisonofMethods To verify the effectiveness of theproposed approach we introduce two existing decision-making methods e first method is the double-referencepoint decision-making method based on the prospect theory[42] e second method is the TOPSIS method [43] In thetwo mentioned methods positive and negative ideal pointsare set as reference points used as the basis of the alternativeevaluation e positive and negative reference points ofattribute j are defined as 1113957rlowast+j and 1113957rlowastminusj respectively Based onthe prospect theory and TOPSIS method the comparisonbetween the attribute value of the alternative and the cor-responding reference points is analyzed and the evaluationvalue of each alternative is obtained In the double-referencepoint decision-making method based on the prospect

theory the maximum and minimum values of public ex-pectation are taken as positive and negative reference points

Public expectations obey the normal distributiontherefore the maximum and minimum values of publicexpectations can be obtained using the three-sigma (3σ)

theorem of normal distribution [44] For attribute j themaximum and minimum values of public expectations areobtained using the 3σ theorem of normal distributiondenoted by μ(rk

j) + 3σ(rkj) and μ(rk

j) minus 3σ(rkj) respectively

According to Definitions 1 and 2 the conversion values oflinguistic variables or uncertain linguistic variables are in therange [0 1] In the conversion crisp valuesrsquo distribution oflinguistic variables or uncertain linguistic variables in orderto ensure the maximum and minimum values we meet therange requirements in this type of distribution we define themaximum and minimum values as follows ifμ(rk

j ) + 3σ(rkj )gt 1 we set maximum value as 1 if

μ(rkj ) minus 3σ(rk

j )lt 0 we set minimum value as 0 For the otherscenarios the maximum and minimum values are set asμ(rk

j ) + 3σ(rkj ) and μ(rk

j) minus 3σ(rkj ) To satisfy the calculation

need of prospect value we set the positive and negativereference points of the attribute as interval numbers withequal upper and lower limits expressed as 1113957rlowast+j [(μ(rk

j) +

3σ(rkj)) (μ(rk

j) + 3σ(rkj))] and 1113957rlowastminusj [(μ(rk

j ) minus 3σ(rkj))

(μ(rkj ) minus 3σ(rk

j ))] In the TOPSISmethod we set the positiveand negative ideal points according to the maximum andminimum values of each attribute in all alternatives Forexample the positive and negative reference points of at-tribute j are respectively expressed as1113957rlowast+j [maxi12MyiU

j maxi12MyiUj ] and

1113957rlowastminusj [mini12MyiLj mini12MyiL

j ] e alternativeranking results of different decision methods are listed inTable 9

According to Table 9 the alternative ranking resultscorresponding to the existing methods proposed are

Table 8 Normalized attribute reference points and attribute values for different alternatives

AlternativeAverage distance between the stationand the population gathering area

along the line (m)

Estimated constructiontime of the project

(months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

1 [10000 10000] [00458 07137] [00318 02734] [03243 10000]2 [06182 06182] [06183 10000] [04943 10000] [10000 10000]3 [09532 09532] [03321 07137] [00318 04179] [03243 10000]Attributereference points [00000 01922] [00000 03798] [00000 04353] [00000 06486]

Table 7 Results of public expectation reference points for different attributes

AttributesAverage distance between the stationand the population gathering area

along the line (m)

Estimatedconstruction time ofthe project (months)

Construction-enclosedpublic areas (m2)

Dust and sandtreatmenteffect

(qualitative index)

Statistical distribution Normal distribution N (4460 74412) Normal distributionN (6949 8482)

Normal distribution N(28990 28992)

Normal distributionN (075 0152)

Comprehensive ambiguityof the expectations 1234 199 100621 006

Attribute reference point [43367 45834] [6750 7148] [2798379 2999621] [069 081]

10 Mathematical Problems in Engineering

inconsistent with the results obtained in this studyemainreason for the inconsistency of decision results is that thereference point setting in each decision method is different

e setting principle of reference points for the twoexisting methods is that reference points are obtainedaccording to the public expectation or the attribute value of thealternative e setting principle for the reference points usedin the two existing methods is relatively simple but it does notconsider the characteristics of public expectations For deci-sion-making activities in major public affairs public expec-tations must be fully considered to ensure the effectiveness ofthe decision-making results erefore public expectationsmust be considered when setting reference points

Based on the above analysis we develop the settingprinciple for reference points by considering public ex-pectations First the information on public expectations isdiscrete and has the characteristics of a normal distributionthus the mean value of public expectations is taken as thestandard setting for the reference point so that the referencepoint can effectively reflect the expectations of the publicgroup Second due to the uncertainty of the expectationevaluation by public individuals the expected value isusually expressed in an interval number which reflects theambiguity of public expectations e comprehensive ex-pected ambiguity of the public is used to represent theuncertain characteristics of public expectations e refer-ence point is set by the mean value of public expectationsand the comprehensive expected ambiguity of the public tobetter reflect the actual situation of public expectation ex-pression e results of the reference points are presented asinterval numbers e value of the reference points reflectsthe expectations of the public group and considers theuncertainty of the public evaluation

If an attribute value in interval form falls within therange of the corresponding reference points in whole or inpart we consider that part of the attribute value that fallswithin the range of the reference points exactly meetspublic expectations which means that the prospect valuefor this part of the attribute is zero erefore when theprospect value is calculated based on the interval reference

point the inclusion or cross relationship between theattribute interval and corresponding reference point in-terval should be considered For example Table 8 indicatesthat the normalized values of attribute 3 for alternatives 1and 3 are p13 [00318 02734] and p33 [00318

04179] respectively Moreover p13 and p33 are not equaland the normalized reference point for attribute 3 isq3 [00000 04353] It is apparent that p13 and p33 areentirely within the interval range of q3 We can affirm thatthe prospect values of attribute 3 for alternatives 1 and 3are both zero (v13 v33 0) In addition we use equations(7) and (8) and Table 8 to calculate the prospect values ofp13 and p33 we can also obtain the same result If wechoose the double-reference point decision-makingmethod to calculate the prospect value of p13 and p33 theresults of the prospect value for p13 and p33 arev13 minus 07329 and v33 minus 06732 which is different fromthe result of the proposed method erefore a differentsetting principle for the reference point leads to differentdecision-making results explaining why the sorting resultsare different in Table 9 erefore if we need to set ref-erence points for decision-making we must set themaccording to the actual situation for the decision problemand decision requirements to guarantee the effectiveness ofthe decision results

52 Sensitivity Analysis of the Reference Point Interval RangeAccording to the content above the comprehensive ambi-guity of public expectations is related to the interval range ofattribute reference points affecting the decision-makingresultse interval adjustment coefficient of reference pointθ (0le θle 1) is introduced in the expression of referencepoints to study the relationship between the value range ofthe reference points and decision results further e ref-erence point of attribute j is defined as1113957rlowastj [1113957rlowastLj 1113957rlowastUj ] [μ(rk

j ) minus θcj μ(rkj) + θcj] e compre-

hensive ambiguity of public expectations for each attribute cj

is known When the values of θ are 0 025 05 075 and 1the corresponding value range of reference points changes

Table 9 Alternative ranking results of different decision methods

Methods Attributereference points Decision results Ranking of

the alternatives

Double-reference point decision-makingmethod based on prospect theory

1113957rlowast+1 [22277 22277] 1113957r

lowast+2 [4405 4405]

1113957rlowast+3 [2029300 2029300] 1113957r

lowast+4 [100 100] V1 minus 01738

V2 minus 00983

V3 minus 01654

V2 gtV3 gtV11113957rlowastminus1 [66923 66923] 1113957r

lowastminus2 [9493 9493]

1113957rlowastminus3 [3768700 3768700] 1113957r

lowastminus4 [000 000]

TOPSIS

1113957rlowast+1 [33000 33000] 1113957r

lowast+2 [6100 6100]

1113957rlowast+3 [2537300 2537300] 1113957r

lowast+4 [0875 0875] V1 05719

V2 04772

V3 05564

V1 gtV3 gtV21113957rlowastminus1 [37900 37900] 1113957r

lowastminus2 [7100 7100]

1113957rlowastminus3 [2984900 2984900] 1113957r

lowastminus4 [075 075]

e proposed method 1113957rlowast1 [43367 45834] 1113957r

lowast2 [6750 7148]

1113957rlowast3 [2798379 2999621] 1113957r

lowast4 [069 081]

V1 03994

V2 04211

V3 03816

V2 gtV1 gtV3

Mathematical Problems in Engineering 11

accordingly e decision results for different conditions ofreference points are presented in Table 10

As listed in Table 10 when the interval adjustmentcoefficient of the reference points increases gradually theinterval range of the reference points for different attributesis also constantly enlarged so the alternative ranking resultsare not the same As observed in Table 10 if θ isin [0 05] thesorting result is V2 gtV3 gtV1 Moreover if θ isin [075 1] thesorting result is V2 gtV1 gtV3 In addition the alternativeranking results in Table 10 with the continuous expansion ofthe interval for reference points indicate that the compre-hensive prospect value of each alternative also constantlydecreases In addition the differentiation degree of thecomprehensive prospect value between alternatives alsogradually decreases erefore to ensure the effectiveness ofsetting attribute reference points we must consider theinfluence of the interval range of the reference points for thedecision results to reduce the difficulty of decision-makingand improve the accuracy of decision results

6 Conclusion

is paper proposed a new decision-making approach for amixed multiattribute decision-making problem with un-known attribute weights e advantages of this approachare summarized as follows First the reference point for eachattribute is set based on the distribution and comprehensiveambiguity of public expectations making the attributereference points better reflect the publicrsquos expected groupopinions and expectation uncertainty e effectiveness ofthe decision result is guaranteed Second in solving theattribute weights the attribute importance given by thepublic is used to determine the value range of the attributeweights so that the weighting results are in accordance withpublic opinions making the results of the attribute weightmore acceptable en the exact values of the attributeweights are determined to maximize the attribute infor-mation deviation of all the alternatives improving thediscrimination of alternatives ird each alternative is

evaluated based on the prospect theory to satisfy the cal-culation needs of the prospect valuee different expressionforms of the attribute information and reference point areunified and normalization is performed e operationabove can eliminate the influence of the expression form anddimension on decision-making making the decision op-erations smoother

e proposed approach also has certain limitationsFirst in the actual decision-making process the decision-making scenarios in some decision-making problems arenot static Changes in the decision-making scenarios easilycause decision risk which negatively affects the decisionactivity Our proposed approach does not consider dy-namic decision scenarios so the proposed approach mustbe further expanded and improved making it suitable forpublic-participation decision-making problems underchanging scenarios Second according to the settingprinciple of reference points the interval range of the at-tribute reference points affects the decision-making resultse interval range of the attribute reference points is largeand the discrimination of the evaluation results for dif-ferent alternatives is less obvious Because the intervalrange of attribute reference points is set by the compre-hensive ambiguity of public expectations if the dispersiondegree of public expectations is overly high (ie thecomprehensive ambiguity of public expectations is overlyhigh) the interval range of attribute reference points is alsoset over wide corresponding values is outcome may leadto a lack of differentiation in the decision results ereforeto ensure the effectiveness of the decision-making resultsreasonable interval ranges must be set for attribute refer-ence points according to the characteristics of public ex-pectation information which is also worth an in-depthstudy

Data Availability

e data used to support the findings of this study are in-cluded within the manuscript

Table 10 Decision results for different conditions of reference points

θ Reference points Comprehensive prospect value Alternatives ranking

0 1113957rlowast1 [44600 44600] 1113957r

lowast2 [6949 6949]

1113957rlowast3 [2899000 2899000] 1113957r

lowast4 [075 075]

V1 04365

V2 06769

V3 04699V2 gtV3 gtV1

025 1113957rlowast1 [44292 44909] 1113957r

lowast2 [6900 7000]

1113957rlowast3 [2873845 2924155] 1113957r

lowast4 [074 077]

V1 04138

V2 06068

V3 04442V2 gtV3 gtV1

050 1113957rlowast1 [43983 45217] 1113957r

lowast2 [6851 7050]

1113957rlowast3 [2848690 2949311] 1113957r

lowast4 [072 078]

V1 04104

V2 05425

V3 04231V2 gtV3 gtV1

075 1113957rlowast1 [43675 45526] 1113957r

lowast2 [6801 7099]

1113957rlowast3 [2823534 2974466] 1113957r

lowast4 [071 080]

V1 04141

V2 04827

V3 04066V2 gtV1 gtV3

1 1113957rlowast1 [43367 45834] 1113957r

lowast2 [6750 7148]

1113957rlowast3 [2798379 2999621] 1113957r

lowast4 [069 081]

V1 03994

V2 04211

V3 03816V2 gtV1 gtV3

12 Mathematical Problems in Engineering

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is research was funded by the National Natural ScienceFoundation of China (no 71902058) Natural ScienceFoundation of Hunan Province (nos 2018JJ3617 and2021JJ41088) Social Science Foundation of Hunan Province(no 19YBQ113) Scientific Research Foundation of HunanEducation Department (nos 18B484 19A06 and 20B155)and Social Science Achievements Appraisal CommitteeFoundation of Hunan Provincial Department (noXSP21YBC218)

References

[1] S Song Z Guo and X Wang ldquoRetracted article the cor-relation between social transformation economic risk andinternet public opinionrdquo Behaviour amp Information Technol-ogy vol 40 no 7 pp 723ndash733 2020

[2] W Chen F Tu and P Zheng ldquoA transnational networkedpublic sphere of air pollution analysis of a Twitter network ofPM25 from the risk society perspectiverdquo InformationCommunication amp Society vol 20 no 7 pp 1005ndash1023 2017

[3] X Yao J He and C Bao ldquoPublic participation modes inChinarsquos environmental impact assessment process an ana-lytical framework based on participation extent and conflictlevelrdquo Environmental Impact Assessment Review vol 84p 106400 2020

[4] X Zhang J G Xu and Y Ju ldquoPublic participation in NIMBYrisk mitigation A discourse zoning approach in the Chinesecontextrdquo Land Use Policy vol 77 pp 559ndash575 2018

[5] T Webler and S Tuler ldquoFour decades of public participationin risk decision makingrdquo Risk Analysis vol 41 no 3pp 503ndash518 2021

[6] J P Voszlig and N Amelung ldquoInnovating public participationmethods Technoscientization and reflexive engagementrdquoSocial Studies of Science vol 46 no 5 pp 749ndash772 2016

[7] A M Rıos B Benito and F Bastida ldquoFactors explainingpublic participation in the central government budget pro-cessrdquo Australian Journal of Public Administration vol 76no 1 pp 48ndash64 2017

[8] Y Zhou L Hou Y Yang H-Y Chong and S Moon ldquoAcomparative review and framework development on publicparticipation for decision-making in Chinese public projectsrdquoEnvironmental Impact Assessment Review vol 75 pp 79ndash872019

[9] T H You J Zhang and Z P Fan ldquoMulti-attribute onlinereview decision making method based on sentiment analysisand evidence theoryrdquo Journal of Systems amp Managementvol 28 no 3 pp 536ndash544 2019 in Chinese

[10] R L Charney T Rebmann P Dalawari and A EndrizalldquoPublic expectations of hospitals to provide resources andservices to the uninjured during disasters A qualitativestudyrdquo Health Security vol 14 no 6 pp 389ndash396 2016

[11] A Yildiz E Ayyildiz A Taskin Gumus and C Ozkan ldquoAframework to prioritize the public expectations from watertreatment plants based on trapezoidal type-2 fuzzy ahpmethodrdquo Environmental Management vol 67 no 3pp 439ndash448 2020

[12] Z Zhang Y Gao and Z L Li ldquoConsensus reaching for socialnetwork group decision making by considering leadershipand bounded confidencerdquo Knowledge-Based Systems vol 204pp 1ndash12 2020

[13] J J Zhu Z Z Ma H H Wang and Y Chen ldquoRisk decision-making method using interval numbers and its applicationbased on the prospect value with multiple reference pointsrdquoInformation Sciences vol 12 no 3 pp 385-386 2017

[14] X Li and X Chen ldquoValue determination method based onmultiple reference points under a trapezoidal intuitionisticfuzzy environmentrdquo Applied Soft Computing vol 63pp 39ndash49 2018

[15] J Gao Z Xu and H Liao ldquoA dynamic reference pointmethod for emergency response under hesitant probabilisticfuzzy environmentrdquo International Journal of Fuzzy Systemsvol 19 no 5 pp 1261ndash1278 2017

[16] C H Li W Li M J Li et al ldquoTarget-oriented model andapproach for attribute value evaluation with multiple refer-ence pointsrdquo Chinese Journal of Management Science vol 25no 7 pp 163ndash175 2017 in Chinese

[17] E Mastrocinque F J Ramırez A Honrubia-Escribano andD T Pham ldquoAn AHP-based multi-criteria model for sus-tainable supply chain development in the renewable energysectorrdquo Expert Systems with Applications vol 150 pp 1ndash172020

[18] L Chen Z Li and X Deng ldquoEmergency alternative evalu-ation under group decision makers a new method based onentropy weight and dematelrdquo International Journal of SystemsScience vol 51 no 3 pp 570ndash583 2020

[19] X KWang Y TWang J QWang P F Cheng and L Li ldquoATODIM-PROMETHEE II based multi-criteria group decisionmaking method for risk evaluation of water resource carryingcapacity under probabilistic linguistic Z-number circum-stancesrdquo Mathematics vol 8 no 7 p 1190 2020

[20] Y Liu Y Dong H Liang F Chiclana and E Herrera-Viedma ldquoMultiple attribute strategic weight manipulationwith minimum cost in a group decision making context withinterval attribute weights informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 49 no 10pp 1981ndash1992 2018

[21] F Meng C Tan and X Chen ldquoAn approach to Atanassovrsquosinterval-valued intuitionistic fuzzy multi-attribute decisionmaking based on prospect theoryrdquo International Journal ofComputational Intelligence Systems vol 8 no 3 pp 591ndash6052015

[22] N Zarbakhshnia Y Wu K Govindan and H Soleimani ldquoAnovel hybrid multiple attribute decision-making approach foroutsourcing sustainable reverse logisticsrdquo Journal of CleanerProduction vol 242 Article ID 118461 2020

[23] J J H Liou Y C Chuang E K Zavadskas and G H TzengldquoData-driven hybrid multiple attribute decision-makingmodel for green supplier evaluation and performance im-provementrdquo Journal of Cleaner Production vol 241 ArticleID 118321 2019

[24] Y H Pan and X L Geng ldquoHybrid multiple attribute decisionmaking approach based on Mo-RVIKORrdquo Chinese Journal ofManagement Science vol 27 no 12 pp 143ndash151 2019

[25] M Akram and D Shumaiza ldquoMulti-criteria decision makingbased on q-rung orthopair fuzzy promethee approachrdquo Ira-nian Journal of Fuzzy Systems vol 18 no 5 pp 107ndash127 2021

[26] C Jana G Muhiuddin and M Pal ldquoMulti-criteria decisionmaking approach based on SVTrN Dombi aggregationfunctionsrdquo Artificial Intelligence Review vol 54 no 5pp 3685ndash3723 2021

Mathematical Problems in Engineering 13

[27] C Jana G Muhiuddin and M Pal ldquoMultiple-attribute de-cision making problems based on SVTNH methodsrdquo Journalof Ambient Intelligence and Humanized Computing vol 11no 9 pp 3717ndash3733 2020

[28] C Jana GMuhiuddin andM Pal ldquoSomeDombi aggregationof Q -rung orthopair fuzzy numbers in multiple-attributedecision makingrdquo International Journal of Intelligent Systemsvol 34 no 12 pp 3220ndash3240 2019

[29] G Shahzadi G Muhiuddin M Arif Butt and A AshrafldquoHamacher interactive hybrid weighted averaging operatorsunder fermatean fuzzy numbersrdquo Journal of Mathematicsvol 2021 no 10 17 pages Article ID 5556017 2021

[30] K Zhang J Zhan and X Wang ldquoTOPSIS-WAA methodbased on a covering-based fuzzy rough set An application torating problemrdquo Information Sciences vol 539 pp 397ndash4212020

[31] S F Huang ldquoUsing Linguistic VIKOR and fuzzy cognitivemaps to select virtual reality games development projectrdquoMathematics vol 9 no 11 p 1253 2021

[32] M Akram A Luqman and C Kahraman ldquoHesitant py-thagorean fuzzy ELECTRE-II method for multi-criteria de-cision-making problemsrdquo Applied Soft Computing vol 108Article ID 107479 2021

[33] C Erdin and G Ozkaya ldquoTurkeyrsquos 2023 energy strategies andinvestment opportunities for renewable energy sources Siteselection based on ELECTRErdquo Sustainability vol 11 no 7Article ID 2136 2019

[34] T Wang H Li X Zhou D Liu and B Huang ldquoree-waydecision based on third-generation prospect theory with Z-numbersrdquo Information Sciences vol 569 pp 13ndash38 2021

[35] Z J Du S M Yu H Y Luo and X D Lin ldquoConsensusconvergence in large-group social network environmentcoordination between trust relationship and opinion simi-larityrdquo Knowledge-Based Systems vol 217 Article ID 1068282021

[36] J Wu F Chiclana H Fujita and E Herrera-Viedma ldquoAvisual interaction consensus model for social network groupdecision making with trust propagationrdquo Knowledge-BasedSystems vol 122 pp 39ndash50 2017

[37] L X Chen and N F Luo ldquoPythagorean fuzzy multi-criteriadecision-making based on prospect theoryrdquo Systems Engi-neering Keory amp Practice vol 40 no 3 pp 726ndash735 2020 inChinese

[38] Z S Xu Linguistic Decision Making Springer-Verlag BerlinGermany 2012

[39] R E Moore Methods and Applications of Interval AnalysisPrentice-Hall Hoboken NY USA 1979

[40] R R Ren W W Li M Zhao and X Li ldquoA large groupdecision making method based on public evaluationrdquoManagement Review vol 30 no 10 pp 238ndash247 2018

[41] L Wang Y M Wang and L Martınez ldquoA group decisionmethod based on prospect theory for emergency situationsrdquoInformation Sciences vol 418-419 pp 119ndash135 2017

[42] Y Wu C Xu and T Zhang ldquoEvaluation of renewable powersources using a fuzzy MCDM based on cumulative prospecttheory a case in Chinardquo Energy vol 147 pp 1227ndash1239 2018

[43] M M Salih B B Zaidan A A Zaidan and M A AhmedldquoSurvey on fuzzy TOPSIS state-of-the-art between 2007 and2017rdquo Computers amp Operations Research vol 104 pp 207ndash227 2019

[44] H Xiao Y Zhang X Liu H Yin P Liu and D C Liu ldquoArapid ultrasound vascular disease screening method usingPauTa Criterionrdquo Journal of Physics Conference Seriesvol 1769 Article ID 012009 2021

14 Mathematical Problems in Engineering

given attribute importance so they choose to express theiropinions using uncertain linguistic variables According tothe expression characteristics of the public Definitions 1 and2 can be used to convert the public evaluation values fromlinguistic or uncertain linguistic variables into intervalnumbers If the number of individuals participating in theimportance evaluation of attribute j is ζj the importanceevaluation value given by individual k over attribute j isεk

j k 1 2 ζj Based on Definitions 1 and 2 εkj can be

converted into the interval type bkj [bkL

j bkUj ]

k 1 2 ζj We take the average value of the importanceevaluation of the public on attribute j as the comprehensiveimportance evaluation value of attribute j (ie bj [bL

j bUj ]

where bLj (1ζj) 1113936

ζj

k1 bkLk bU

j (1ζj) 1113936ζj

k1 bkUk 0le bL

j lebU

j le 1)Second the value range of the attribute weights ωj is

determined e value range of the attribute weights ωj isassumed to be ωj isin [ωL

j ωUj ] Based on the public com-

prehensive evaluation value of the importance over attributej the upper and lower limits of ωj are determined

ωLj

bLj

bLj + 1113936

Nminus 1e1enejb

Ue

ωUj

bUj

bUj + 1113936

Nminus 1e1enejb

Le

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)

Theorem 1 For ωj isin [ωLj ωU

j ] ωj must exist that meets theconstraints of 0leωj le 1 and 1113936

Nj1 ωj 1

Proof Because 0le bLj le (bL

j + 1113936Nminus 1e1enejb

Ue ) and

0le bUj le (bU

j + 1113936Nminus 1e1enejb

Le ) 0le (bL

j (bLj + 1113936

Nminus 1e1enejb

Ue ))le 1 and

0le (bUj (bU

j + 1113936Nminus 1e1enejb

Le ))le 1 us it can be deduced that

0leωj le 1 As 0le bLj le bU

j le 1 and ωLj (bL

j (bLj + 1113936

Nminus 1e1enej

bUe ))le (bL

j 1113936Nj1 bL

j ) and it can be deduced that (1113936Nj1 ω

Lj

1113936Nj1[bL

j (bLj + 1113936

Nminus 1e1enejb

Ue )])le 1113936

Nj1(bL

j 1113936Nj1 bL

j ) 1 Simi-larly it is deduced that 1113936

Nj1 ω

Uj ge 1 As the value of ωj is

continuous within the range [ωLj ωU

j ] and 0leωLj leωU

j le 1 so1113936

Nj1 ωj 1 must existird we determine the attribute weights An optimi-

zation model is constructed to solve the attribute weights tomaximize the dispersion of attributes on all alternatives

maxψ ωj1113872 1113873 1

M1113944

M

i11113944

N

j1

pLij minus 1113957p

Lj

11138681113868111386811138681113868

11138681113868111386811138681113868 + pUij minus 1113957p

Uj

11138681113868111386811138681113868

11138681113868111386811138681113868

2⎛⎝ ⎞⎠ωj

st

ωLj leωj leω

Uj

1113944

N

j1ωj 1

1113957pj 1113957pLj 1113957p

Uj1113960 1113961

1M

1113944

M

i1p

Lij

1M

1113944

M

i1p

Uij

⎡⎣ ⎤⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)

Table 1 Equations to calculate Gij and Fij

No e relationship between pij and qj e loss Fij e profit Gij

1 pUij lt qL

j 05(pLij + pU

ij) minus qLj 0

2 qUj ltpL

ij 0 05(pLij + pU

ij) minus qUj

3 pLij lt qL

j lepUij lt qU

j 05(pLij minus qL

j ) 0

4 qLj ltpL

ij le qUj ltpU

ij 0 05(pUij minus qU

j )

5 pLij lt qL

j lt qUj ltpU

ij 05(pLij minus qL

j ) 05(pUij minus qU

j )

6 qLj lepL

ij ltpUij le qU

j 0 0

6 Mathematical Problems in Engineering

Theorem 2 Model (10) must have an optimal solution

Proof Under the constraint ωj isin [ωLj ωU

j ] there must be areasonable value of ωj that satisfies 1113936

Nj1 ωj 1 so the

feasible domain of the attribute weight is a nonempty setIn addition 0lepij le 1 and 0lepj le 1 so it is easy to de-duce that 0le ((|pL

ij minus pLj | + |pU

ij minus pUj |)2)le 1 As known

0leωj le 1 1113936Nj1 ωj 1 thus it can be deduced

that 0le 1113936Nj1((|pL

ij minus pLj | + |pU

ij minus pUj |)2)ωj le 1 and 0le

ψ(ωj) (1M) 1113936Mi1 1113936

Nj1 ((|pL

ij minus 1113957pLj |+ |pU

ij minus 1113957pUj |)2)ωj le 1

As ψ(ωj) is a bounded continuous function the constraintcondition of the attribute weight is a bounded closed set soModel (10) must have an optimal solution

To sum up the specific steps of the proposed decision-making approach are as follows

Step 1 e public expected values of attributes withvarious expressions are converted into crisp numbersStep 2e distribution of the public expected values isdetermined on all attributes according to the publicexpected opinion in the crisp number type Next theattribute reference points are obtained based on thedistribution mean of the public expectation andcomprehensive ambiguity of public expected valuesStep 3 e attribute value and reference points arenormalized e attribute prospect value of each al-ternative is calculated based on the prospect theoryStep 4 e value range of the attribute weights isdetermined using equation (9) and Model (10) to de-termine the attribute weightStep 5 e comprehensive prospect values of differentalternatives are obtained using equation (11) to realizethe ranking of alternative alternatives

Vi 1113944N

j1ωjvij (11)

4 Case Analysis

We take a subway construction project as an example toverify the rationality and effectiveness of the method pro-posed in this paper A provincial capital city plans to extendthe No 2 subway line to the west for which three alter-natives can be selected e extension of the subway line willmake public transportation more convenient for residentsalong the line However the subway construction will take along time to be completed occupy a large amount of publicspace and generate substantial dust which interferes withthe daily life of the surrounding residents According to theconstruction requirements and characteristics of the subwayline the organizers of the decision-making activity selectedfour attributes to evaluate the alternatives the averagedistance between the stations and densely populated areasalong the line (g1 units m cost-based index) estimated

construction time (g2 units month cost-based index)enclosed public area for construction (g3 units m

2 cost-based index) and dust and sand treatment effect (g4qualitative index profit-based index) Among these alter-natives g1 is expressed in crisp numbers g2 and g3 areexpressed in interval numbers and g4 is expressed in lin-guistic or uncertain linguistic variables e conversionstandard between the dust and sand treatment effect andlinguistic variables is presented in Table 2 and the attributevalues of different alternatives are listed in Table 3

e primary public group affected by the constructionand operation of the subway is urban residents ereforeduring the decision-making process the opinions of thepublic group directly affected by the subway must be fullyconsidered Various media-driven methods were used topublicize the project to enable the public to understand theactual subway project situation better e public couldexpress their opinions on the subway project using differentmethods such as online platforms telephone and mailquestionnaires e public provides two pieces of evaluationinformation based on their opinions of the attributes theirexpectations and the importance evaluation value econversion standard between the evaluation values of theimportance and linguistic variables is given in Table 2

When the public opinion survey was finished the or-ganizers of the public opinion survey identified and countedthe public individuals who effectively participated estatistical results of the public expectations are listed inTable 4 and the statistical results of the public importanceevaluation values of different attributes are presented inTable 5 Numerous individuals participated in the surveyeffectively Due to the space limitations of the article we onlypresent the partial statistical results of the public opinions inTables 4 and 5

e original public expected values of the attributes inTable 4 were processed First the public expected values ofthe attributes were converted into interval numbers Nextthe comprehensive ambiguity of the expected values of theattributes was calculated which was shown in Table 4 enthe public expected values of the attributes in intervalnumbers were converted into crisp numbers in Definition 3the details of the public expected values for different attri-butes in the form of crisp numbers are shown in Table 6

Finally based on the relevant content in Table 6 thedistribution of public expected values for various attributeswas examined e distribution of public expected values onvarious attributes was examined e fittings of the distri-butions are illustrated in Figures 1ndash4 According to thestatistical distribution results and comprehensive ambiguityof the public expected values the reference point of eachattribute was determined as presented in Table 7

Equations (3)ndash(6) were used to normalize the attributereference points and attribute values of different alternativese normalized attribute reference points and alternativeattribute values are listed in Table 8

Next equations (7) and (8) were used to calculate theprospect values of the attributes and the prospect profit-lossmatrix of the attributes is expressed

Mathematical Problems in Engineering 7

Table 2 Conversion standards between dust and sand treatment (attribute importance) and linguistic variables

Dust and sand treatmenteffect (attribute importance)

Extremely poor (canbe ignored)

Terribly poor(extremely

unimportant)

Very poor (veryunimportant) Poor (unimportant) Fair

(good)

Linguistic variable s0 s1 s2 s3 s4

Dust and sand treatmenteffect (attribute importance) Good (important) Very good (very

important)Extremely good

(extremely important)Perfect (maximum

importance) mdash

Linguistic variable s5 s6 s7 s8 mdash

Table 3 Attribute values of various alternatives

AlternativesAverage distance between the station

and the population gatheringarea along the line (m)

Estimated constructiontime of the

project (months)

Construction enclosedpublic areas (m2)

Dust and sandtreatment

effect (qualitative index)1 330 [64 71] [28732 29849] [S6 S7]2 379 [61 65] [25373 27711] S73 336 [64 68] [28064 29849] [S6 S7]

Table 4 Statistics of public expected values for different attributes

Average distance between thestation and the population

gathering area along the line (m)

Estimated construction time ofthe project (months)

Construction-enclosed publicareas (m2)

Dust and sand treatment effect(qualitative index)

Number of effective participants Number of effective participants Number of effective participants Number of effective participants5472 6070 5518 5491

No Expectedvalues Ambiguity No Expected

value Ambiguity No Expectedvalue Ambiguity No Expected

value Ambiguity

1 [453 499] 23 1 [55 58] 15 1 [3052830789] 1305 1 [S4 S5] 00625

2 [377 377] 0 2 [64 66] 1 2 [2621229284] 1536 2 [S5 S5] 0

5471 [346 352] 3 6069 [67 74] 35 5517 [3004933846] 18985 5490 [S5 S6] 00625

5472 [338 374] 18 6070 [70 78] 4 5518 [2817531285] 1555 5491 [S6 S6] 0

Table 5 Public evaluation results of the importance for different attributes

Average distance betweenthe station and the

population gathering areaalong the line (m)

Estimated constructiontime of the project

(months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

Number of effectiveparticipants

Number of effectiveparticipants

Number of effectiveparticipants

Number of effectiveparticipants

4186 4182 4186 4185No Importance No Importance No Importance No Importance1 [S3 S4] 1 [S1 S2] 1 [S3 S3] 1 [S3 S5]2 [S3 S4] 2 [S1 S2] 2 [S3 S3] 2 [S3 S5]

4185 [S3 S6] 4181 [S2 S2] 4185 [S1 S4] 4184 [S4 S5]4186 [S3 S6] 4182 [S0 S1] 4186 [S1 S4] 4185 [S4 S5]

8 Mathematical Problems in Engineering

Table 6 e public expected values for different attributes in the form of crisp numbers

Average distance betweenthe station and the

population gathering areaalong the line (m)

Estimated construction timeof the project (months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

No Expected value No Expected value No Expected value No Expected value1 476 1 56 1 306585 1 056252 377 2 65 2 27748 2 05000

5471 349 6069 705 5517 319475 5490 068755472 356 6070 74 5518 29730 5491 07500

250 300 350 400 450 500 550 600 6500

20

40

60

80

100

120

140

160

180

Freq

uenc

y

Frequency

Public expectation of attribute 1 (m)

Normal distribution fitting curve

Figure 1 Fitting curve for public expected values of attribute 1

55 60 65 70 75 80 85 900

50

100

150

200

250

300

350

Freq

uenc

y

FrequencyNormal distribution fitting curve

Public expectation of attribute 2 (months)

Figure 2 Fitting curve for public expected values of attribute 2

22 24 26 28 3 32 34 36 380

20

40

60

80

100

120

140

Freq

uenc

y

Frequency

times104

Normal distribution fitting curve

Public expectation of attribute 3 (m2)

Figure 3 Fitting curve for public expected values of attribute 3

05 06 07 08 09 1 11 12Public expectation of attribute 4 (qualitative index)

0

500

1000

1500

2000

2500

Freq

uenc

y

FrequencyNormal distribution fitting curve

Figure 4 Fitting curve for public expected values of attribute 4

Mathematical Problems in Engineering 9

VM

08287 02070 00000 02164

04719 04752 03587 03983

07864 02070 00000 02164

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (12)

e value range of the attribute weights can be deter-mined using equation (9) based on the conversion betweenlinguistic variables and interval numbers ω1 isin [024 055]

ω2 isin [007 026] ω3 isin [010 034]ω4 isin [017 044] Attri-bute weights can be determined using Model (10)ω1 042ω2 007ω3 034ω4 017

Based on the attribute prospect value and attributeweights the comprehensive prospect value of each alter-native can be obtained using equation (11) whereV1 03994 V2 04211 and V3 03816 e ranking ofthree alternatives is V2 gtV1 gtV3 erefore V2 is the op-timal alternative

5 Comparison of Methods andSensitivity Analysis

51ComparisonofMethods To verify the effectiveness of theproposed approach we introduce two existing decision-making methods e first method is the double-referencepoint decision-making method based on the prospect theory[42] e second method is the TOPSIS method [43] In thetwo mentioned methods positive and negative ideal pointsare set as reference points used as the basis of the alternativeevaluation e positive and negative reference points ofattribute j are defined as 1113957rlowast+j and 1113957rlowastminusj respectively Based onthe prospect theory and TOPSIS method the comparisonbetween the attribute value of the alternative and the cor-responding reference points is analyzed and the evaluationvalue of each alternative is obtained In the double-referencepoint decision-making method based on the prospect

theory the maximum and minimum values of public ex-pectation are taken as positive and negative reference points

Public expectations obey the normal distributiontherefore the maximum and minimum values of publicexpectations can be obtained using the three-sigma (3σ)

theorem of normal distribution [44] For attribute j themaximum and minimum values of public expectations areobtained using the 3σ theorem of normal distributiondenoted by μ(rk

j) + 3σ(rkj) and μ(rk

j) minus 3σ(rkj) respectively

According to Definitions 1 and 2 the conversion values oflinguistic variables or uncertain linguistic variables are in therange [0 1] In the conversion crisp valuesrsquo distribution oflinguistic variables or uncertain linguistic variables in orderto ensure the maximum and minimum values we meet therange requirements in this type of distribution we define themaximum and minimum values as follows ifμ(rk

j ) + 3σ(rkj )gt 1 we set maximum value as 1 if

μ(rkj ) minus 3σ(rk

j )lt 0 we set minimum value as 0 For the otherscenarios the maximum and minimum values are set asμ(rk

j ) + 3σ(rkj ) and μ(rk

j) minus 3σ(rkj ) To satisfy the calculation

need of prospect value we set the positive and negativereference points of the attribute as interval numbers withequal upper and lower limits expressed as 1113957rlowast+j [(μ(rk

j) +

3σ(rkj)) (μ(rk

j) + 3σ(rkj))] and 1113957rlowastminusj [(μ(rk

j ) minus 3σ(rkj))

(μ(rkj ) minus 3σ(rk

j ))] In the TOPSISmethod we set the positiveand negative ideal points according to the maximum andminimum values of each attribute in all alternatives Forexample the positive and negative reference points of at-tribute j are respectively expressed as1113957rlowast+j [maxi12MyiU

j maxi12MyiUj ] and

1113957rlowastminusj [mini12MyiLj mini12MyiL

j ] e alternativeranking results of different decision methods are listed inTable 9

According to Table 9 the alternative ranking resultscorresponding to the existing methods proposed are

Table 8 Normalized attribute reference points and attribute values for different alternatives

AlternativeAverage distance between the stationand the population gathering area

along the line (m)

Estimated constructiontime of the project

(months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

1 [10000 10000] [00458 07137] [00318 02734] [03243 10000]2 [06182 06182] [06183 10000] [04943 10000] [10000 10000]3 [09532 09532] [03321 07137] [00318 04179] [03243 10000]Attributereference points [00000 01922] [00000 03798] [00000 04353] [00000 06486]

Table 7 Results of public expectation reference points for different attributes

AttributesAverage distance between the stationand the population gathering area

along the line (m)

Estimatedconstruction time ofthe project (months)

Construction-enclosedpublic areas (m2)

Dust and sandtreatmenteffect

(qualitative index)

Statistical distribution Normal distribution N (4460 74412) Normal distributionN (6949 8482)

Normal distribution N(28990 28992)

Normal distributionN (075 0152)

Comprehensive ambiguityof the expectations 1234 199 100621 006

Attribute reference point [43367 45834] [6750 7148] [2798379 2999621] [069 081]

10 Mathematical Problems in Engineering

inconsistent with the results obtained in this studyemainreason for the inconsistency of decision results is that thereference point setting in each decision method is different

e setting principle of reference points for the twoexisting methods is that reference points are obtainedaccording to the public expectation or the attribute value of thealternative e setting principle for the reference points usedin the two existing methods is relatively simple but it does notconsider the characteristics of public expectations For deci-sion-making activities in major public affairs public expec-tations must be fully considered to ensure the effectiveness ofthe decision-making results erefore public expectationsmust be considered when setting reference points

Based on the above analysis we develop the settingprinciple for reference points by considering public ex-pectations First the information on public expectations isdiscrete and has the characteristics of a normal distributionthus the mean value of public expectations is taken as thestandard setting for the reference point so that the referencepoint can effectively reflect the expectations of the publicgroup Second due to the uncertainty of the expectationevaluation by public individuals the expected value isusually expressed in an interval number which reflects theambiguity of public expectations e comprehensive ex-pected ambiguity of the public is used to represent theuncertain characteristics of public expectations e refer-ence point is set by the mean value of public expectationsand the comprehensive expected ambiguity of the public tobetter reflect the actual situation of public expectation ex-pression e results of the reference points are presented asinterval numbers e value of the reference points reflectsthe expectations of the public group and considers theuncertainty of the public evaluation

If an attribute value in interval form falls within therange of the corresponding reference points in whole or inpart we consider that part of the attribute value that fallswithin the range of the reference points exactly meetspublic expectations which means that the prospect valuefor this part of the attribute is zero erefore when theprospect value is calculated based on the interval reference

point the inclusion or cross relationship between theattribute interval and corresponding reference point in-terval should be considered For example Table 8 indicatesthat the normalized values of attribute 3 for alternatives 1and 3 are p13 [00318 02734] and p33 [00318

04179] respectively Moreover p13 and p33 are not equaland the normalized reference point for attribute 3 isq3 [00000 04353] It is apparent that p13 and p33 areentirely within the interval range of q3 We can affirm thatthe prospect values of attribute 3 for alternatives 1 and 3are both zero (v13 v33 0) In addition we use equations(7) and (8) and Table 8 to calculate the prospect values ofp13 and p33 we can also obtain the same result If wechoose the double-reference point decision-makingmethod to calculate the prospect value of p13 and p33 theresults of the prospect value for p13 and p33 arev13 minus 07329 and v33 minus 06732 which is different fromthe result of the proposed method erefore a differentsetting principle for the reference point leads to differentdecision-making results explaining why the sorting resultsare different in Table 9 erefore if we need to set ref-erence points for decision-making we must set themaccording to the actual situation for the decision problemand decision requirements to guarantee the effectiveness ofthe decision results

52 Sensitivity Analysis of the Reference Point Interval RangeAccording to the content above the comprehensive ambi-guity of public expectations is related to the interval range ofattribute reference points affecting the decision-makingresultse interval adjustment coefficient of reference pointθ (0le θle 1) is introduced in the expression of referencepoints to study the relationship between the value range ofthe reference points and decision results further e ref-erence point of attribute j is defined as1113957rlowastj [1113957rlowastLj 1113957rlowastUj ] [μ(rk

j ) minus θcj μ(rkj) + θcj] e compre-

hensive ambiguity of public expectations for each attribute cj

is known When the values of θ are 0 025 05 075 and 1the corresponding value range of reference points changes

Table 9 Alternative ranking results of different decision methods

Methods Attributereference points Decision results Ranking of

the alternatives

Double-reference point decision-makingmethod based on prospect theory

1113957rlowast+1 [22277 22277] 1113957r

lowast+2 [4405 4405]

1113957rlowast+3 [2029300 2029300] 1113957r

lowast+4 [100 100] V1 minus 01738

V2 minus 00983

V3 minus 01654

V2 gtV3 gtV11113957rlowastminus1 [66923 66923] 1113957r

lowastminus2 [9493 9493]

1113957rlowastminus3 [3768700 3768700] 1113957r

lowastminus4 [000 000]

TOPSIS

1113957rlowast+1 [33000 33000] 1113957r

lowast+2 [6100 6100]

1113957rlowast+3 [2537300 2537300] 1113957r

lowast+4 [0875 0875] V1 05719

V2 04772

V3 05564

V1 gtV3 gtV21113957rlowastminus1 [37900 37900] 1113957r

lowastminus2 [7100 7100]

1113957rlowastminus3 [2984900 2984900] 1113957r

lowastminus4 [075 075]

e proposed method 1113957rlowast1 [43367 45834] 1113957r

lowast2 [6750 7148]

1113957rlowast3 [2798379 2999621] 1113957r

lowast4 [069 081]

V1 03994

V2 04211

V3 03816

V2 gtV1 gtV3

Mathematical Problems in Engineering 11

accordingly e decision results for different conditions ofreference points are presented in Table 10

As listed in Table 10 when the interval adjustmentcoefficient of the reference points increases gradually theinterval range of the reference points for different attributesis also constantly enlarged so the alternative ranking resultsare not the same As observed in Table 10 if θ isin [0 05] thesorting result is V2 gtV3 gtV1 Moreover if θ isin [075 1] thesorting result is V2 gtV1 gtV3 In addition the alternativeranking results in Table 10 with the continuous expansion ofthe interval for reference points indicate that the compre-hensive prospect value of each alternative also constantlydecreases In addition the differentiation degree of thecomprehensive prospect value between alternatives alsogradually decreases erefore to ensure the effectiveness ofsetting attribute reference points we must consider theinfluence of the interval range of the reference points for thedecision results to reduce the difficulty of decision-makingand improve the accuracy of decision results

6 Conclusion

is paper proposed a new decision-making approach for amixed multiattribute decision-making problem with un-known attribute weights e advantages of this approachare summarized as follows First the reference point for eachattribute is set based on the distribution and comprehensiveambiguity of public expectations making the attributereference points better reflect the publicrsquos expected groupopinions and expectation uncertainty e effectiveness ofthe decision result is guaranteed Second in solving theattribute weights the attribute importance given by thepublic is used to determine the value range of the attributeweights so that the weighting results are in accordance withpublic opinions making the results of the attribute weightmore acceptable en the exact values of the attributeweights are determined to maximize the attribute infor-mation deviation of all the alternatives improving thediscrimination of alternatives ird each alternative is

evaluated based on the prospect theory to satisfy the cal-culation needs of the prospect valuee different expressionforms of the attribute information and reference point areunified and normalization is performed e operationabove can eliminate the influence of the expression form anddimension on decision-making making the decision op-erations smoother

e proposed approach also has certain limitationsFirst in the actual decision-making process the decision-making scenarios in some decision-making problems arenot static Changes in the decision-making scenarios easilycause decision risk which negatively affects the decisionactivity Our proposed approach does not consider dy-namic decision scenarios so the proposed approach mustbe further expanded and improved making it suitable forpublic-participation decision-making problems underchanging scenarios Second according to the settingprinciple of reference points the interval range of the at-tribute reference points affects the decision-making resultse interval range of the attribute reference points is largeand the discrimination of the evaluation results for dif-ferent alternatives is less obvious Because the intervalrange of attribute reference points is set by the compre-hensive ambiguity of public expectations if the dispersiondegree of public expectations is overly high (ie thecomprehensive ambiguity of public expectations is overlyhigh) the interval range of attribute reference points is alsoset over wide corresponding values is outcome may leadto a lack of differentiation in the decision results ereforeto ensure the effectiveness of the decision-making resultsreasonable interval ranges must be set for attribute refer-ence points according to the characteristics of public ex-pectation information which is also worth an in-depthstudy

Data Availability

e data used to support the findings of this study are in-cluded within the manuscript

Table 10 Decision results for different conditions of reference points

θ Reference points Comprehensive prospect value Alternatives ranking

0 1113957rlowast1 [44600 44600] 1113957r

lowast2 [6949 6949]

1113957rlowast3 [2899000 2899000] 1113957r

lowast4 [075 075]

V1 04365

V2 06769

V3 04699V2 gtV3 gtV1

025 1113957rlowast1 [44292 44909] 1113957r

lowast2 [6900 7000]

1113957rlowast3 [2873845 2924155] 1113957r

lowast4 [074 077]

V1 04138

V2 06068

V3 04442V2 gtV3 gtV1

050 1113957rlowast1 [43983 45217] 1113957r

lowast2 [6851 7050]

1113957rlowast3 [2848690 2949311] 1113957r

lowast4 [072 078]

V1 04104

V2 05425

V3 04231V2 gtV3 gtV1

075 1113957rlowast1 [43675 45526] 1113957r

lowast2 [6801 7099]

1113957rlowast3 [2823534 2974466] 1113957r

lowast4 [071 080]

V1 04141

V2 04827

V3 04066V2 gtV1 gtV3

1 1113957rlowast1 [43367 45834] 1113957r

lowast2 [6750 7148]

1113957rlowast3 [2798379 2999621] 1113957r

lowast4 [069 081]

V1 03994

V2 04211

V3 03816V2 gtV1 gtV3

12 Mathematical Problems in Engineering

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is research was funded by the National Natural ScienceFoundation of China (no 71902058) Natural ScienceFoundation of Hunan Province (nos 2018JJ3617 and2021JJ41088) Social Science Foundation of Hunan Province(no 19YBQ113) Scientific Research Foundation of HunanEducation Department (nos 18B484 19A06 and 20B155)and Social Science Achievements Appraisal CommitteeFoundation of Hunan Provincial Department (noXSP21YBC218)

References

[1] S Song Z Guo and X Wang ldquoRetracted article the cor-relation between social transformation economic risk andinternet public opinionrdquo Behaviour amp Information Technol-ogy vol 40 no 7 pp 723ndash733 2020

[2] W Chen F Tu and P Zheng ldquoA transnational networkedpublic sphere of air pollution analysis of a Twitter network ofPM25 from the risk society perspectiverdquo InformationCommunication amp Society vol 20 no 7 pp 1005ndash1023 2017

[3] X Yao J He and C Bao ldquoPublic participation modes inChinarsquos environmental impact assessment process an ana-lytical framework based on participation extent and conflictlevelrdquo Environmental Impact Assessment Review vol 84p 106400 2020

[4] X Zhang J G Xu and Y Ju ldquoPublic participation in NIMBYrisk mitigation A discourse zoning approach in the Chinesecontextrdquo Land Use Policy vol 77 pp 559ndash575 2018

[5] T Webler and S Tuler ldquoFour decades of public participationin risk decision makingrdquo Risk Analysis vol 41 no 3pp 503ndash518 2021

[6] J P Voszlig and N Amelung ldquoInnovating public participationmethods Technoscientization and reflexive engagementrdquoSocial Studies of Science vol 46 no 5 pp 749ndash772 2016

[7] A M Rıos B Benito and F Bastida ldquoFactors explainingpublic participation in the central government budget pro-cessrdquo Australian Journal of Public Administration vol 76no 1 pp 48ndash64 2017

[8] Y Zhou L Hou Y Yang H-Y Chong and S Moon ldquoAcomparative review and framework development on publicparticipation for decision-making in Chinese public projectsrdquoEnvironmental Impact Assessment Review vol 75 pp 79ndash872019

[9] T H You J Zhang and Z P Fan ldquoMulti-attribute onlinereview decision making method based on sentiment analysisand evidence theoryrdquo Journal of Systems amp Managementvol 28 no 3 pp 536ndash544 2019 in Chinese

[10] R L Charney T Rebmann P Dalawari and A EndrizalldquoPublic expectations of hospitals to provide resources andservices to the uninjured during disasters A qualitativestudyrdquo Health Security vol 14 no 6 pp 389ndash396 2016

[11] A Yildiz E Ayyildiz A Taskin Gumus and C Ozkan ldquoAframework to prioritize the public expectations from watertreatment plants based on trapezoidal type-2 fuzzy ahpmethodrdquo Environmental Management vol 67 no 3pp 439ndash448 2020

[12] Z Zhang Y Gao and Z L Li ldquoConsensus reaching for socialnetwork group decision making by considering leadershipand bounded confidencerdquo Knowledge-Based Systems vol 204pp 1ndash12 2020

[13] J J Zhu Z Z Ma H H Wang and Y Chen ldquoRisk decision-making method using interval numbers and its applicationbased on the prospect value with multiple reference pointsrdquoInformation Sciences vol 12 no 3 pp 385-386 2017

[14] X Li and X Chen ldquoValue determination method based onmultiple reference points under a trapezoidal intuitionisticfuzzy environmentrdquo Applied Soft Computing vol 63pp 39ndash49 2018

[15] J Gao Z Xu and H Liao ldquoA dynamic reference pointmethod for emergency response under hesitant probabilisticfuzzy environmentrdquo International Journal of Fuzzy Systemsvol 19 no 5 pp 1261ndash1278 2017

[16] C H Li W Li M J Li et al ldquoTarget-oriented model andapproach for attribute value evaluation with multiple refer-ence pointsrdquo Chinese Journal of Management Science vol 25no 7 pp 163ndash175 2017 in Chinese

[17] E Mastrocinque F J Ramırez A Honrubia-Escribano andD T Pham ldquoAn AHP-based multi-criteria model for sus-tainable supply chain development in the renewable energysectorrdquo Expert Systems with Applications vol 150 pp 1ndash172020

[18] L Chen Z Li and X Deng ldquoEmergency alternative evalu-ation under group decision makers a new method based onentropy weight and dematelrdquo International Journal of SystemsScience vol 51 no 3 pp 570ndash583 2020

[19] X KWang Y TWang J QWang P F Cheng and L Li ldquoATODIM-PROMETHEE II based multi-criteria group decisionmaking method for risk evaluation of water resource carryingcapacity under probabilistic linguistic Z-number circum-stancesrdquo Mathematics vol 8 no 7 p 1190 2020

[20] Y Liu Y Dong H Liang F Chiclana and E Herrera-Viedma ldquoMultiple attribute strategic weight manipulationwith minimum cost in a group decision making context withinterval attribute weights informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 49 no 10pp 1981ndash1992 2018

[21] F Meng C Tan and X Chen ldquoAn approach to Atanassovrsquosinterval-valued intuitionistic fuzzy multi-attribute decisionmaking based on prospect theoryrdquo International Journal ofComputational Intelligence Systems vol 8 no 3 pp 591ndash6052015

[22] N Zarbakhshnia Y Wu K Govindan and H Soleimani ldquoAnovel hybrid multiple attribute decision-making approach foroutsourcing sustainable reverse logisticsrdquo Journal of CleanerProduction vol 242 Article ID 118461 2020

[23] J J H Liou Y C Chuang E K Zavadskas and G H TzengldquoData-driven hybrid multiple attribute decision-makingmodel for green supplier evaluation and performance im-provementrdquo Journal of Cleaner Production vol 241 ArticleID 118321 2019

[24] Y H Pan and X L Geng ldquoHybrid multiple attribute decisionmaking approach based on Mo-RVIKORrdquo Chinese Journal ofManagement Science vol 27 no 12 pp 143ndash151 2019

[25] M Akram and D Shumaiza ldquoMulti-criteria decision makingbased on q-rung orthopair fuzzy promethee approachrdquo Ira-nian Journal of Fuzzy Systems vol 18 no 5 pp 107ndash127 2021

[26] C Jana G Muhiuddin and M Pal ldquoMulti-criteria decisionmaking approach based on SVTrN Dombi aggregationfunctionsrdquo Artificial Intelligence Review vol 54 no 5pp 3685ndash3723 2021

Mathematical Problems in Engineering 13

[27] C Jana G Muhiuddin and M Pal ldquoMultiple-attribute de-cision making problems based on SVTNH methodsrdquo Journalof Ambient Intelligence and Humanized Computing vol 11no 9 pp 3717ndash3733 2020

[28] C Jana GMuhiuddin andM Pal ldquoSomeDombi aggregationof Q -rung orthopair fuzzy numbers in multiple-attributedecision makingrdquo International Journal of Intelligent Systemsvol 34 no 12 pp 3220ndash3240 2019

[29] G Shahzadi G Muhiuddin M Arif Butt and A AshrafldquoHamacher interactive hybrid weighted averaging operatorsunder fermatean fuzzy numbersrdquo Journal of Mathematicsvol 2021 no 10 17 pages Article ID 5556017 2021

[30] K Zhang J Zhan and X Wang ldquoTOPSIS-WAA methodbased on a covering-based fuzzy rough set An application torating problemrdquo Information Sciences vol 539 pp 397ndash4212020

[31] S F Huang ldquoUsing Linguistic VIKOR and fuzzy cognitivemaps to select virtual reality games development projectrdquoMathematics vol 9 no 11 p 1253 2021

[32] M Akram A Luqman and C Kahraman ldquoHesitant py-thagorean fuzzy ELECTRE-II method for multi-criteria de-cision-making problemsrdquo Applied Soft Computing vol 108Article ID 107479 2021

[33] C Erdin and G Ozkaya ldquoTurkeyrsquos 2023 energy strategies andinvestment opportunities for renewable energy sources Siteselection based on ELECTRErdquo Sustainability vol 11 no 7Article ID 2136 2019

[34] T Wang H Li X Zhou D Liu and B Huang ldquoree-waydecision based on third-generation prospect theory with Z-numbersrdquo Information Sciences vol 569 pp 13ndash38 2021

[35] Z J Du S M Yu H Y Luo and X D Lin ldquoConsensusconvergence in large-group social network environmentcoordination between trust relationship and opinion simi-larityrdquo Knowledge-Based Systems vol 217 Article ID 1068282021

[36] J Wu F Chiclana H Fujita and E Herrera-Viedma ldquoAvisual interaction consensus model for social network groupdecision making with trust propagationrdquo Knowledge-BasedSystems vol 122 pp 39ndash50 2017

[37] L X Chen and N F Luo ldquoPythagorean fuzzy multi-criteriadecision-making based on prospect theoryrdquo Systems Engi-neering Keory amp Practice vol 40 no 3 pp 726ndash735 2020 inChinese

[38] Z S Xu Linguistic Decision Making Springer-Verlag BerlinGermany 2012

[39] R E Moore Methods and Applications of Interval AnalysisPrentice-Hall Hoboken NY USA 1979

[40] R R Ren W W Li M Zhao and X Li ldquoA large groupdecision making method based on public evaluationrdquoManagement Review vol 30 no 10 pp 238ndash247 2018

[41] L Wang Y M Wang and L Martınez ldquoA group decisionmethod based on prospect theory for emergency situationsrdquoInformation Sciences vol 418-419 pp 119ndash135 2017

[42] Y Wu C Xu and T Zhang ldquoEvaluation of renewable powersources using a fuzzy MCDM based on cumulative prospecttheory a case in Chinardquo Energy vol 147 pp 1227ndash1239 2018

[43] M M Salih B B Zaidan A A Zaidan and M A AhmedldquoSurvey on fuzzy TOPSIS state-of-the-art between 2007 and2017rdquo Computers amp Operations Research vol 104 pp 207ndash227 2019

[44] H Xiao Y Zhang X Liu H Yin P Liu and D C Liu ldquoArapid ultrasound vascular disease screening method usingPauTa Criterionrdquo Journal of Physics Conference Seriesvol 1769 Article ID 012009 2021

14 Mathematical Problems in Engineering

Theorem 2 Model (10) must have an optimal solution

Proof Under the constraint ωj isin [ωLj ωU

j ] there must be areasonable value of ωj that satisfies 1113936

Nj1 ωj 1 so the

feasible domain of the attribute weight is a nonempty setIn addition 0lepij le 1 and 0lepj le 1 so it is easy to de-duce that 0le ((|pL

ij minus pLj | + |pU

ij minus pUj |)2)le 1 As known

0leωj le 1 1113936Nj1 ωj 1 thus it can be deduced

that 0le 1113936Nj1((|pL

ij minus pLj | + |pU

ij minus pUj |)2)ωj le 1 and 0le

ψ(ωj) (1M) 1113936Mi1 1113936

Nj1 ((|pL

ij minus 1113957pLj |+ |pU

ij minus 1113957pUj |)2)ωj le 1

As ψ(ωj) is a bounded continuous function the constraintcondition of the attribute weight is a bounded closed set soModel (10) must have an optimal solution

To sum up the specific steps of the proposed decision-making approach are as follows

Step 1 e public expected values of attributes withvarious expressions are converted into crisp numbersStep 2e distribution of the public expected values isdetermined on all attributes according to the publicexpected opinion in the crisp number type Next theattribute reference points are obtained based on thedistribution mean of the public expectation andcomprehensive ambiguity of public expected valuesStep 3 e attribute value and reference points arenormalized e attribute prospect value of each al-ternative is calculated based on the prospect theoryStep 4 e value range of the attribute weights isdetermined using equation (9) and Model (10) to de-termine the attribute weightStep 5 e comprehensive prospect values of differentalternatives are obtained using equation (11) to realizethe ranking of alternative alternatives

Vi 1113944N

j1ωjvij (11)

4 Case Analysis

We take a subway construction project as an example toverify the rationality and effectiveness of the method pro-posed in this paper A provincial capital city plans to extendthe No 2 subway line to the west for which three alter-natives can be selected e extension of the subway line willmake public transportation more convenient for residentsalong the line However the subway construction will take along time to be completed occupy a large amount of publicspace and generate substantial dust which interferes withthe daily life of the surrounding residents According to theconstruction requirements and characteristics of the subwayline the organizers of the decision-making activity selectedfour attributes to evaluate the alternatives the averagedistance between the stations and densely populated areasalong the line (g1 units m cost-based index) estimated

construction time (g2 units month cost-based index)enclosed public area for construction (g3 units m

2 cost-based index) and dust and sand treatment effect (g4qualitative index profit-based index) Among these alter-natives g1 is expressed in crisp numbers g2 and g3 areexpressed in interval numbers and g4 is expressed in lin-guistic or uncertain linguistic variables e conversionstandard between the dust and sand treatment effect andlinguistic variables is presented in Table 2 and the attributevalues of different alternatives are listed in Table 3

e primary public group affected by the constructionand operation of the subway is urban residents ereforeduring the decision-making process the opinions of thepublic group directly affected by the subway must be fullyconsidered Various media-driven methods were used topublicize the project to enable the public to understand theactual subway project situation better e public couldexpress their opinions on the subway project using differentmethods such as online platforms telephone and mailquestionnaires e public provides two pieces of evaluationinformation based on their opinions of the attributes theirexpectations and the importance evaluation value econversion standard between the evaluation values of theimportance and linguistic variables is given in Table 2

When the public opinion survey was finished the or-ganizers of the public opinion survey identified and countedthe public individuals who effectively participated estatistical results of the public expectations are listed inTable 4 and the statistical results of the public importanceevaluation values of different attributes are presented inTable 5 Numerous individuals participated in the surveyeffectively Due to the space limitations of the article we onlypresent the partial statistical results of the public opinions inTables 4 and 5

e original public expected values of the attributes inTable 4 were processed First the public expected values ofthe attributes were converted into interval numbers Nextthe comprehensive ambiguity of the expected values of theattributes was calculated which was shown in Table 4 enthe public expected values of the attributes in intervalnumbers were converted into crisp numbers in Definition 3the details of the public expected values for different attri-butes in the form of crisp numbers are shown in Table 6

Finally based on the relevant content in Table 6 thedistribution of public expected values for various attributeswas examined e distribution of public expected values onvarious attributes was examined e fittings of the distri-butions are illustrated in Figures 1ndash4 According to thestatistical distribution results and comprehensive ambiguityof the public expected values the reference point of eachattribute was determined as presented in Table 7

Equations (3)ndash(6) were used to normalize the attributereference points and attribute values of different alternativese normalized attribute reference points and alternativeattribute values are listed in Table 8

Next equations (7) and (8) were used to calculate theprospect values of the attributes and the prospect profit-lossmatrix of the attributes is expressed

Mathematical Problems in Engineering 7

Table 2 Conversion standards between dust and sand treatment (attribute importance) and linguistic variables

Dust and sand treatmenteffect (attribute importance)

Extremely poor (canbe ignored)

Terribly poor(extremely

unimportant)

Very poor (veryunimportant) Poor (unimportant) Fair

(good)

Linguistic variable s0 s1 s2 s3 s4

Dust and sand treatmenteffect (attribute importance) Good (important) Very good (very

important)Extremely good

(extremely important)Perfect (maximum

importance) mdash

Linguistic variable s5 s6 s7 s8 mdash

Table 3 Attribute values of various alternatives

AlternativesAverage distance between the station

and the population gatheringarea along the line (m)

Estimated constructiontime of the

project (months)

Construction enclosedpublic areas (m2)

Dust and sandtreatment

effect (qualitative index)1 330 [64 71] [28732 29849] [S6 S7]2 379 [61 65] [25373 27711] S73 336 [64 68] [28064 29849] [S6 S7]

Table 4 Statistics of public expected values for different attributes

Average distance between thestation and the population

gathering area along the line (m)

Estimated construction time ofthe project (months)

Construction-enclosed publicareas (m2)

Dust and sand treatment effect(qualitative index)

Number of effective participants Number of effective participants Number of effective participants Number of effective participants5472 6070 5518 5491

No Expectedvalues Ambiguity No Expected

value Ambiguity No Expectedvalue Ambiguity No Expected

value Ambiguity

1 [453 499] 23 1 [55 58] 15 1 [3052830789] 1305 1 [S4 S5] 00625

2 [377 377] 0 2 [64 66] 1 2 [2621229284] 1536 2 [S5 S5] 0

5471 [346 352] 3 6069 [67 74] 35 5517 [3004933846] 18985 5490 [S5 S6] 00625

5472 [338 374] 18 6070 [70 78] 4 5518 [2817531285] 1555 5491 [S6 S6] 0

Table 5 Public evaluation results of the importance for different attributes

Average distance betweenthe station and the

population gathering areaalong the line (m)

Estimated constructiontime of the project

(months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

Number of effectiveparticipants

Number of effectiveparticipants

Number of effectiveparticipants

Number of effectiveparticipants

4186 4182 4186 4185No Importance No Importance No Importance No Importance1 [S3 S4] 1 [S1 S2] 1 [S3 S3] 1 [S3 S5]2 [S3 S4] 2 [S1 S2] 2 [S3 S3] 2 [S3 S5]

4185 [S3 S6] 4181 [S2 S2] 4185 [S1 S4] 4184 [S4 S5]4186 [S3 S6] 4182 [S0 S1] 4186 [S1 S4] 4185 [S4 S5]

8 Mathematical Problems in Engineering

Table 6 e public expected values for different attributes in the form of crisp numbers

Average distance betweenthe station and the

population gathering areaalong the line (m)

Estimated construction timeof the project (months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

No Expected value No Expected value No Expected value No Expected value1 476 1 56 1 306585 1 056252 377 2 65 2 27748 2 05000

5471 349 6069 705 5517 319475 5490 068755472 356 6070 74 5518 29730 5491 07500

250 300 350 400 450 500 550 600 6500

20

40

60

80

100

120

140

160

180

Freq

uenc

y

Frequency

Public expectation of attribute 1 (m)

Normal distribution fitting curve

Figure 1 Fitting curve for public expected values of attribute 1

55 60 65 70 75 80 85 900

50

100

150

200

250

300

350

Freq

uenc

y

FrequencyNormal distribution fitting curve

Public expectation of attribute 2 (months)

Figure 2 Fitting curve for public expected values of attribute 2

22 24 26 28 3 32 34 36 380

20

40

60

80

100

120

140

Freq

uenc

y

Frequency

times104

Normal distribution fitting curve

Public expectation of attribute 3 (m2)

Figure 3 Fitting curve for public expected values of attribute 3

05 06 07 08 09 1 11 12Public expectation of attribute 4 (qualitative index)

0

500

1000

1500

2000

2500

Freq

uenc

y

FrequencyNormal distribution fitting curve

Figure 4 Fitting curve for public expected values of attribute 4

Mathematical Problems in Engineering 9

VM

08287 02070 00000 02164

04719 04752 03587 03983

07864 02070 00000 02164

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (12)

e value range of the attribute weights can be deter-mined using equation (9) based on the conversion betweenlinguistic variables and interval numbers ω1 isin [024 055]

ω2 isin [007 026] ω3 isin [010 034]ω4 isin [017 044] Attri-bute weights can be determined using Model (10)ω1 042ω2 007ω3 034ω4 017

Based on the attribute prospect value and attributeweights the comprehensive prospect value of each alter-native can be obtained using equation (11) whereV1 03994 V2 04211 and V3 03816 e ranking ofthree alternatives is V2 gtV1 gtV3 erefore V2 is the op-timal alternative

5 Comparison of Methods andSensitivity Analysis

51ComparisonofMethods To verify the effectiveness of theproposed approach we introduce two existing decision-making methods e first method is the double-referencepoint decision-making method based on the prospect theory[42] e second method is the TOPSIS method [43] In thetwo mentioned methods positive and negative ideal pointsare set as reference points used as the basis of the alternativeevaluation e positive and negative reference points ofattribute j are defined as 1113957rlowast+j and 1113957rlowastminusj respectively Based onthe prospect theory and TOPSIS method the comparisonbetween the attribute value of the alternative and the cor-responding reference points is analyzed and the evaluationvalue of each alternative is obtained In the double-referencepoint decision-making method based on the prospect

theory the maximum and minimum values of public ex-pectation are taken as positive and negative reference points

Public expectations obey the normal distributiontherefore the maximum and minimum values of publicexpectations can be obtained using the three-sigma (3σ)

theorem of normal distribution [44] For attribute j themaximum and minimum values of public expectations areobtained using the 3σ theorem of normal distributiondenoted by μ(rk

j) + 3σ(rkj) and μ(rk

j) minus 3σ(rkj) respectively

According to Definitions 1 and 2 the conversion values oflinguistic variables or uncertain linguistic variables are in therange [0 1] In the conversion crisp valuesrsquo distribution oflinguistic variables or uncertain linguistic variables in orderto ensure the maximum and minimum values we meet therange requirements in this type of distribution we define themaximum and minimum values as follows ifμ(rk

j ) + 3σ(rkj )gt 1 we set maximum value as 1 if

μ(rkj ) minus 3σ(rk

j )lt 0 we set minimum value as 0 For the otherscenarios the maximum and minimum values are set asμ(rk

j ) + 3σ(rkj ) and μ(rk

j) minus 3σ(rkj ) To satisfy the calculation

need of prospect value we set the positive and negativereference points of the attribute as interval numbers withequal upper and lower limits expressed as 1113957rlowast+j [(μ(rk

j) +

3σ(rkj)) (μ(rk

j) + 3σ(rkj))] and 1113957rlowastminusj [(μ(rk

j ) minus 3σ(rkj))

(μ(rkj ) minus 3σ(rk

j ))] In the TOPSISmethod we set the positiveand negative ideal points according to the maximum andminimum values of each attribute in all alternatives Forexample the positive and negative reference points of at-tribute j are respectively expressed as1113957rlowast+j [maxi12MyiU

j maxi12MyiUj ] and

1113957rlowastminusj [mini12MyiLj mini12MyiL

j ] e alternativeranking results of different decision methods are listed inTable 9

According to Table 9 the alternative ranking resultscorresponding to the existing methods proposed are

Table 8 Normalized attribute reference points and attribute values for different alternatives

AlternativeAverage distance between the stationand the population gathering area

along the line (m)

Estimated constructiontime of the project

(months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

1 [10000 10000] [00458 07137] [00318 02734] [03243 10000]2 [06182 06182] [06183 10000] [04943 10000] [10000 10000]3 [09532 09532] [03321 07137] [00318 04179] [03243 10000]Attributereference points [00000 01922] [00000 03798] [00000 04353] [00000 06486]

Table 7 Results of public expectation reference points for different attributes

AttributesAverage distance between the stationand the population gathering area

along the line (m)

Estimatedconstruction time ofthe project (months)

Construction-enclosedpublic areas (m2)

Dust and sandtreatmenteffect

(qualitative index)

Statistical distribution Normal distribution N (4460 74412) Normal distributionN (6949 8482)

Normal distribution N(28990 28992)

Normal distributionN (075 0152)

Comprehensive ambiguityof the expectations 1234 199 100621 006

Attribute reference point [43367 45834] [6750 7148] [2798379 2999621] [069 081]

10 Mathematical Problems in Engineering

inconsistent with the results obtained in this studyemainreason for the inconsistency of decision results is that thereference point setting in each decision method is different

e setting principle of reference points for the twoexisting methods is that reference points are obtainedaccording to the public expectation or the attribute value of thealternative e setting principle for the reference points usedin the two existing methods is relatively simple but it does notconsider the characteristics of public expectations For deci-sion-making activities in major public affairs public expec-tations must be fully considered to ensure the effectiveness ofthe decision-making results erefore public expectationsmust be considered when setting reference points

Based on the above analysis we develop the settingprinciple for reference points by considering public ex-pectations First the information on public expectations isdiscrete and has the characteristics of a normal distributionthus the mean value of public expectations is taken as thestandard setting for the reference point so that the referencepoint can effectively reflect the expectations of the publicgroup Second due to the uncertainty of the expectationevaluation by public individuals the expected value isusually expressed in an interval number which reflects theambiguity of public expectations e comprehensive ex-pected ambiguity of the public is used to represent theuncertain characteristics of public expectations e refer-ence point is set by the mean value of public expectationsand the comprehensive expected ambiguity of the public tobetter reflect the actual situation of public expectation ex-pression e results of the reference points are presented asinterval numbers e value of the reference points reflectsthe expectations of the public group and considers theuncertainty of the public evaluation

If an attribute value in interval form falls within therange of the corresponding reference points in whole or inpart we consider that part of the attribute value that fallswithin the range of the reference points exactly meetspublic expectations which means that the prospect valuefor this part of the attribute is zero erefore when theprospect value is calculated based on the interval reference

point the inclusion or cross relationship between theattribute interval and corresponding reference point in-terval should be considered For example Table 8 indicatesthat the normalized values of attribute 3 for alternatives 1and 3 are p13 [00318 02734] and p33 [00318

04179] respectively Moreover p13 and p33 are not equaland the normalized reference point for attribute 3 isq3 [00000 04353] It is apparent that p13 and p33 areentirely within the interval range of q3 We can affirm thatthe prospect values of attribute 3 for alternatives 1 and 3are both zero (v13 v33 0) In addition we use equations(7) and (8) and Table 8 to calculate the prospect values ofp13 and p33 we can also obtain the same result If wechoose the double-reference point decision-makingmethod to calculate the prospect value of p13 and p33 theresults of the prospect value for p13 and p33 arev13 minus 07329 and v33 minus 06732 which is different fromthe result of the proposed method erefore a differentsetting principle for the reference point leads to differentdecision-making results explaining why the sorting resultsare different in Table 9 erefore if we need to set ref-erence points for decision-making we must set themaccording to the actual situation for the decision problemand decision requirements to guarantee the effectiveness ofthe decision results

52 Sensitivity Analysis of the Reference Point Interval RangeAccording to the content above the comprehensive ambi-guity of public expectations is related to the interval range ofattribute reference points affecting the decision-makingresultse interval adjustment coefficient of reference pointθ (0le θle 1) is introduced in the expression of referencepoints to study the relationship between the value range ofthe reference points and decision results further e ref-erence point of attribute j is defined as1113957rlowastj [1113957rlowastLj 1113957rlowastUj ] [μ(rk

j ) minus θcj μ(rkj) + θcj] e compre-

hensive ambiguity of public expectations for each attribute cj

is known When the values of θ are 0 025 05 075 and 1the corresponding value range of reference points changes

Table 9 Alternative ranking results of different decision methods

Methods Attributereference points Decision results Ranking of

the alternatives

Double-reference point decision-makingmethod based on prospect theory

1113957rlowast+1 [22277 22277] 1113957r

lowast+2 [4405 4405]

1113957rlowast+3 [2029300 2029300] 1113957r

lowast+4 [100 100] V1 minus 01738

V2 minus 00983

V3 minus 01654

V2 gtV3 gtV11113957rlowastminus1 [66923 66923] 1113957r

lowastminus2 [9493 9493]

1113957rlowastminus3 [3768700 3768700] 1113957r

lowastminus4 [000 000]

TOPSIS

1113957rlowast+1 [33000 33000] 1113957r

lowast+2 [6100 6100]

1113957rlowast+3 [2537300 2537300] 1113957r

lowast+4 [0875 0875] V1 05719

V2 04772

V3 05564

V1 gtV3 gtV21113957rlowastminus1 [37900 37900] 1113957r

lowastminus2 [7100 7100]

1113957rlowastminus3 [2984900 2984900] 1113957r

lowastminus4 [075 075]

e proposed method 1113957rlowast1 [43367 45834] 1113957r

lowast2 [6750 7148]

1113957rlowast3 [2798379 2999621] 1113957r

lowast4 [069 081]

V1 03994

V2 04211

V3 03816

V2 gtV1 gtV3

Mathematical Problems in Engineering 11

accordingly e decision results for different conditions ofreference points are presented in Table 10

As listed in Table 10 when the interval adjustmentcoefficient of the reference points increases gradually theinterval range of the reference points for different attributesis also constantly enlarged so the alternative ranking resultsare not the same As observed in Table 10 if θ isin [0 05] thesorting result is V2 gtV3 gtV1 Moreover if θ isin [075 1] thesorting result is V2 gtV1 gtV3 In addition the alternativeranking results in Table 10 with the continuous expansion ofthe interval for reference points indicate that the compre-hensive prospect value of each alternative also constantlydecreases In addition the differentiation degree of thecomprehensive prospect value between alternatives alsogradually decreases erefore to ensure the effectiveness ofsetting attribute reference points we must consider theinfluence of the interval range of the reference points for thedecision results to reduce the difficulty of decision-makingand improve the accuracy of decision results

6 Conclusion

is paper proposed a new decision-making approach for amixed multiattribute decision-making problem with un-known attribute weights e advantages of this approachare summarized as follows First the reference point for eachattribute is set based on the distribution and comprehensiveambiguity of public expectations making the attributereference points better reflect the publicrsquos expected groupopinions and expectation uncertainty e effectiveness ofthe decision result is guaranteed Second in solving theattribute weights the attribute importance given by thepublic is used to determine the value range of the attributeweights so that the weighting results are in accordance withpublic opinions making the results of the attribute weightmore acceptable en the exact values of the attributeweights are determined to maximize the attribute infor-mation deviation of all the alternatives improving thediscrimination of alternatives ird each alternative is

evaluated based on the prospect theory to satisfy the cal-culation needs of the prospect valuee different expressionforms of the attribute information and reference point areunified and normalization is performed e operationabove can eliminate the influence of the expression form anddimension on decision-making making the decision op-erations smoother

e proposed approach also has certain limitationsFirst in the actual decision-making process the decision-making scenarios in some decision-making problems arenot static Changes in the decision-making scenarios easilycause decision risk which negatively affects the decisionactivity Our proposed approach does not consider dy-namic decision scenarios so the proposed approach mustbe further expanded and improved making it suitable forpublic-participation decision-making problems underchanging scenarios Second according to the settingprinciple of reference points the interval range of the at-tribute reference points affects the decision-making resultse interval range of the attribute reference points is largeand the discrimination of the evaluation results for dif-ferent alternatives is less obvious Because the intervalrange of attribute reference points is set by the compre-hensive ambiguity of public expectations if the dispersiondegree of public expectations is overly high (ie thecomprehensive ambiguity of public expectations is overlyhigh) the interval range of attribute reference points is alsoset over wide corresponding values is outcome may leadto a lack of differentiation in the decision results ereforeto ensure the effectiveness of the decision-making resultsreasonable interval ranges must be set for attribute refer-ence points according to the characteristics of public ex-pectation information which is also worth an in-depthstudy

Data Availability

e data used to support the findings of this study are in-cluded within the manuscript

Table 10 Decision results for different conditions of reference points

θ Reference points Comprehensive prospect value Alternatives ranking

0 1113957rlowast1 [44600 44600] 1113957r

lowast2 [6949 6949]

1113957rlowast3 [2899000 2899000] 1113957r

lowast4 [075 075]

V1 04365

V2 06769

V3 04699V2 gtV3 gtV1

025 1113957rlowast1 [44292 44909] 1113957r

lowast2 [6900 7000]

1113957rlowast3 [2873845 2924155] 1113957r

lowast4 [074 077]

V1 04138

V2 06068

V3 04442V2 gtV3 gtV1

050 1113957rlowast1 [43983 45217] 1113957r

lowast2 [6851 7050]

1113957rlowast3 [2848690 2949311] 1113957r

lowast4 [072 078]

V1 04104

V2 05425

V3 04231V2 gtV3 gtV1

075 1113957rlowast1 [43675 45526] 1113957r

lowast2 [6801 7099]

1113957rlowast3 [2823534 2974466] 1113957r

lowast4 [071 080]

V1 04141

V2 04827

V3 04066V2 gtV1 gtV3

1 1113957rlowast1 [43367 45834] 1113957r

lowast2 [6750 7148]

1113957rlowast3 [2798379 2999621] 1113957r

lowast4 [069 081]

V1 03994

V2 04211

V3 03816V2 gtV1 gtV3

12 Mathematical Problems in Engineering

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is research was funded by the National Natural ScienceFoundation of China (no 71902058) Natural ScienceFoundation of Hunan Province (nos 2018JJ3617 and2021JJ41088) Social Science Foundation of Hunan Province(no 19YBQ113) Scientific Research Foundation of HunanEducation Department (nos 18B484 19A06 and 20B155)and Social Science Achievements Appraisal CommitteeFoundation of Hunan Provincial Department (noXSP21YBC218)

References

[1] S Song Z Guo and X Wang ldquoRetracted article the cor-relation between social transformation economic risk andinternet public opinionrdquo Behaviour amp Information Technol-ogy vol 40 no 7 pp 723ndash733 2020

[2] W Chen F Tu and P Zheng ldquoA transnational networkedpublic sphere of air pollution analysis of a Twitter network ofPM25 from the risk society perspectiverdquo InformationCommunication amp Society vol 20 no 7 pp 1005ndash1023 2017

[3] X Yao J He and C Bao ldquoPublic participation modes inChinarsquos environmental impact assessment process an ana-lytical framework based on participation extent and conflictlevelrdquo Environmental Impact Assessment Review vol 84p 106400 2020

[4] X Zhang J G Xu and Y Ju ldquoPublic participation in NIMBYrisk mitigation A discourse zoning approach in the Chinesecontextrdquo Land Use Policy vol 77 pp 559ndash575 2018

[5] T Webler and S Tuler ldquoFour decades of public participationin risk decision makingrdquo Risk Analysis vol 41 no 3pp 503ndash518 2021

[6] J P Voszlig and N Amelung ldquoInnovating public participationmethods Technoscientization and reflexive engagementrdquoSocial Studies of Science vol 46 no 5 pp 749ndash772 2016

[7] A M Rıos B Benito and F Bastida ldquoFactors explainingpublic participation in the central government budget pro-cessrdquo Australian Journal of Public Administration vol 76no 1 pp 48ndash64 2017

[8] Y Zhou L Hou Y Yang H-Y Chong and S Moon ldquoAcomparative review and framework development on publicparticipation for decision-making in Chinese public projectsrdquoEnvironmental Impact Assessment Review vol 75 pp 79ndash872019

[9] T H You J Zhang and Z P Fan ldquoMulti-attribute onlinereview decision making method based on sentiment analysisand evidence theoryrdquo Journal of Systems amp Managementvol 28 no 3 pp 536ndash544 2019 in Chinese

[10] R L Charney T Rebmann P Dalawari and A EndrizalldquoPublic expectations of hospitals to provide resources andservices to the uninjured during disasters A qualitativestudyrdquo Health Security vol 14 no 6 pp 389ndash396 2016

[11] A Yildiz E Ayyildiz A Taskin Gumus and C Ozkan ldquoAframework to prioritize the public expectations from watertreatment plants based on trapezoidal type-2 fuzzy ahpmethodrdquo Environmental Management vol 67 no 3pp 439ndash448 2020

[12] Z Zhang Y Gao and Z L Li ldquoConsensus reaching for socialnetwork group decision making by considering leadershipand bounded confidencerdquo Knowledge-Based Systems vol 204pp 1ndash12 2020

[13] J J Zhu Z Z Ma H H Wang and Y Chen ldquoRisk decision-making method using interval numbers and its applicationbased on the prospect value with multiple reference pointsrdquoInformation Sciences vol 12 no 3 pp 385-386 2017

[14] X Li and X Chen ldquoValue determination method based onmultiple reference points under a trapezoidal intuitionisticfuzzy environmentrdquo Applied Soft Computing vol 63pp 39ndash49 2018

[15] J Gao Z Xu and H Liao ldquoA dynamic reference pointmethod for emergency response under hesitant probabilisticfuzzy environmentrdquo International Journal of Fuzzy Systemsvol 19 no 5 pp 1261ndash1278 2017

[16] C H Li W Li M J Li et al ldquoTarget-oriented model andapproach for attribute value evaluation with multiple refer-ence pointsrdquo Chinese Journal of Management Science vol 25no 7 pp 163ndash175 2017 in Chinese

[17] E Mastrocinque F J Ramırez A Honrubia-Escribano andD T Pham ldquoAn AHP-based multi-criteria model for sus-tainable supply chain development in the renewable energysectorrdquo Expert Systems with Applications vol 150 pp 1ndash172020

[18] L Chen Z Li and X Deng ldquoEmergency alternative evalu-ation under group decision makers a new method based onentropy weight and dematelrdquo International Journal of SystemsScience vol 51 no 3 pp 570ndash583 2020

[19] X KWang Y TWang J QWang P F Cheng and L Li ldquoATODIM-PROMETHEE II based multi-criteria group decisionmaking method for risk evaluation of water resource carryingcapacity under probabilistic linguistic Z-number circum-stancesrdquo Mathematics vol 8 no 7 p 1190 2020

[20] Y Liu Y Dong H Liang F Chiclana and E Herrera-Viedma ldquoMultiple attribute strategic weight manipulationwith minimum cost in a group decision making context withinterval attribute weights informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 49 no 10pp 1981ndash1992 2018

[21] F Meng C Tan and X Chen ldquoAn approach to Atanassovrsquosinterval-valued intuitionistic fuzzy multi-attribute decisionmaking based on prospect theoryrdquo International Journal ofComputational Intelligence Systems vol 8 no 3 pp 591ndash6052015

[22] N Zarbakhshnia Y Wu K Govindan and H Soleimani ldquoAnovel hybrid multiple attribute decision-making approach foroutsourcing sustainable reverse logisticsrdquo Journal of CleanerProduction vol 242 Article ID 118461 2020

[23] J J H Liou Y C Chuang E K Zavadskas and G H TzengldquoData-driven hybrid multiple attribute decision-makingmodel for green supplier evaluation and performance im-provementrdquo Journal of Cleaner Production vol 241 ArticleID 118321 2019

[24] Y H Pan and X L Geng ldquoHybrid multiple attribute decisionmaking approach based on Mo-RVIKORrdquo Chinese Journal ofManagement Science vol 27 no 12 pp 143ndash151 2019

[25] M Akram and D Shumaiza ldquoMulti-criteria decision makingbased on q-rung orthopair fuzzy promethee approachrdquo Ira-nian Journal of Fuzzy Systems vol 18 no 5 pp 107ndash127 2021

[26] C Jana G Muhiuddin and M Pal ldquoMulti-criteria decisionmaking approach based on SVTrN Dombi aggregationfunctionsrdquo Artificial Intelligence Review vol 54 no 5pp 3685ndash3723 2021

Mathematical Problems in Engineering 13

[27] C Jana G Muhiuddin and M Pal ldquoMultiple-attribute de-cision making problems based on SVTNH methodsrdquo Journalof Ambient Intelligence and Humanized Computing vol 11no 9 pp 3717ndash3733 2020

[28] C Jana GMuhiuddin andM Pal ldquoSomeDombi aggregationof Q -rung orthopair fuzzy numbers in multiple-attributedecision makingrdquo International Journal of Intelligent Systemsvol 34 no 12 pp 3220ndash3240 2019

[29] G Shahzadi G Muhiuddin M Arif Butt and A AshrafldquoHamacher interactive hybrid weighted averaging operatorsunder fermatean fuzzy numbersrdquo Journal of Mathematicsvol 2021 no 10 17 pages Article ID 5556017 2021

[30] K Zhang J Zhan and X Wang ldquoTOPSIS-WAA methodbased on a covering-based fuzzy rough set An application torating problemrdquo Information Sciences vol 539 pp 397ndash4212020

[31] S F Huang ldquoUsing Linguistic VIKOR and fuzzy cognitivemaps to select virtual reality games development projectrdquoMathematics vol 9 no 11 p 1253 2021

[32] M Akram A Luqman and C Kahraman ldquoHesitant py-thagorean fuzzy ELECTRE-II method for multi-criteria de-cision-making problemsrdquo Applied Soft Computing vol 108Article ID 107479 2021

[33] C Erdin and G Ozkaya ldquoTurkeyrsquos 2023 energy strategies andinvestment opportunities for renewable energy sources Siteselection based on ELECTRErdquo Sustainability vol 11 no 7Article ID 2136 2019

[34] T Wang H Li X Zhou D Liu and B Huang ldquoree-waydecision based on third-generation prospect theory with Z-numbersrdquo Information Sciences vol 569 pp 13ndash38 2021

[35] Z J Du S M Yu H Y Luo and X D Lin ldquoConsensusconvergence in large-group social network environmentcoordination between trust relationship and opinion simi-larityrdquo Knowledge-Based Systems vol 217 Article ID 1068282021

[36] J Wu F Chiclana H Fujita and E Herrera-Viedma ldquoAvisual interaction consensus model for social network groupdecision making with trust propagationrdquo Knowledge-BasedSystems vol 122 pp 39ndash50 2017

[37] L X Chen and N F Luo ldquoPythagorean fuzzy multi-criteriadecision-making based on prospect theoryrdquo Systems Engi-neering Keory amp Practice vol 40 no 3 pp 726ndash735 2020 inChinese

[38] Z S Xu Linguistic Decision Making Springer-Verlag BerlinGermany 2012

[39] R E Moore Methods and Applications of Interval AnalysisPrentice-Hall Hoboken NY USA 1979

[40] R R Ren W W Li M Zhao and X Li ldquoA large groupdecision making method based on public evaluationrdquoManagement Review vol 30 no 10 pp 238ndash247 2018

[41] L Wang Y M Wang and L Martınez ldquoA group decisionmethod based on prospect theory for emergency situationsrdquoInformation Sciences vol 418-419 pp 119ndash135 2017

[42] Y Wu C Xu and T Zhang ldquoEvaluation of renewable powersources using a fuzzy MCDM based on cumulative prospecttheory a case in Chinardquo Energy vol 147 pp 1227ndash1239 2018

[43] M M Salih B B Zaidan A A Zaidan and M A AhmedldquoSurvey on fuzzy TOPSIS state-of-the-art between 2007 and2017rdquo Computers amp Operations Research vol 104 pp 207ndash227 2019

[44] H Xiao Y Zhang X Liu H Yin P Liu and D C Liu ldquoArapid ultrasound vascular disease screening method usingPauTa Criterionrdquo Journal of Physics Conference Seriesvol 1769 Article ID 012009 2021

14 Mathematical Problems in Engineering

Table 2 Conversion standards between dust and sand treatment (attribute importance) and linguistic variables

Dust and sand treatmenteffect (attribute importance)

Extremely poor (canbe ignored)

Terribly poor(extremely

unimportant)

Very poor (veryunimportant) Poor (unimportant) Fair

(good)

Linguistic variable s0 s1 s2 s3 s4

Dust and sand treatmenteffect (attribute importance) Good (important) Very good (very

important)Extremely good

(extremely important)Perfect (maximum

importance) mdash

Linguistic variable s5 s6 s7 s8 mdash

Table 3 Attribute values of various alternatives

AlternativesAverage distance between the station

and the population gatheringarea along the line (m)

Estimated constructiontime of the

project (months)

Construction enclosedpublic areas (m2)

Dust and sandtreatment

effect (qualitative index)1 330 [64 71] [28732 29849] [S6 S7]2 379 [61 65] [25373 27711] S73 336 [64 68] [28064 29849] [S6 S7]

Table 4 Statistics of public expected values for different attributes

Average distance between thestation and the population

gathering area along the line (m)

Estimated construction time ofthe project (months)

Construction-enclosed publicareas (m2)

Dust and sand treatment effect(qualitative index)

Number of effective participants Number of effective participants Number of effective participants Number of effective participants5472 6070 5518 5491

No Expectedvalues Ambiguity No Expected

value Ambiguity No Expectedvalue Ambiguity No Expected

value Ambiguity

1 [453 499] 23 1 [55 58] 15 1 [3052830789] 1305 1 [S4 S5] 00625

2 [377 377] 0 2 [64 66] 1 2 [2621229284] 1536 2 [S5 S5] 0

5471 [346 352] 3 6069 [67 74] 35 5517 [3004933846] 18985 5490 [S5 S6] 00625

5472 [338 374] 18 6070 [70 78] 4 5518 [2817531285] 1555 5491 [S6 S6] 0

Table 5 Public evaluation results of the importance for different attributes

Average distance betweenthe station and the

population gathering areaalong the line (m)

Estimated constructiontime of the project

(months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

Number of effectiveparticipants

Number of effectiveparticipants

Number of effectiveparticipants

Number of effectiveparticipants

4186 4182 4186 4185No Importance No Importance No Importance No Importance1 [S3 S4] 1 [S1 S2] 1 [S3 S3] 1 [S3 S5]2 [S3 S4] 2 [S1 S2] 2 [S3 S3] 2 [S3 S5]

4185 [S3 S6] 4181 [S2 S2] 4185 [S1 S4] 4184 [S4 S5]4186 [S3 S6] 4182 [S0 S1] 4186 [S1 S4] 4185 [S4 S5]

8 Mathematical Problems in Engineering

Table 6 e public expected values for different attributes in the form of crisp numbers

Average distance betweenthe station and the

population gathering areaalong the line (m)

Estimated construction timeof the project (months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

No Expected value No Expected value No Expected value No Expected value1 476 1 56 1 306585 1 056252 377 2 65 2 27748 2 05000

5471 349 6069 705 5517 319475 5490 068755472 356 6070 74 5518 29730 5491 07500

250 300 350 400 450 500 550 600 6500

20

40

60

80

100

120

140

160

180

Freq

uenc

y

Frequency

Public expectation of attribute 1 (m)

Normal distribution fitting curve

Figure 1 Fitting curve for public expected values of attribute 1

55 60 65 70 75 80 85 900

50

100

150

200

250

300

350

Freq

uenc

y

FrequencyNormal distribution fitting curve

Public expectation of attribute 2 (months)

Figure 2 Fitting curve for public expected values of attribute 2

22 24 26 28 3 32 34 36 380

20

40

60

80

100

120

140

Freq

uenc

y

Frequency

times104

Normal distribution fitting curve

Public expectation of attribute 3 (m2)

Figure 3 Fitting curve for public expected values of attribute 3

05 06 07 08 09 1 11 12Public expectation of attribute 4 (qualitative index)

0

500

1000

1500

2000

2500

Freq

uenc

y

FrequencyNormal distribution fitting curve

Figure 4 Fitting curve for public expected values of attribute 4

Mathematical Problems in Engineering 9

VM

08287 02070 00000 02164

04719 04752 03587 03983

07864 02070 00000 02164

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (12)

e value range of the attribute weights can be deter-mined using equation (9) based on the conversion betweenlinguistic variables and interval numbers ω1 isin [024 055]

ω2 isin [007 026] ω3 isin [010 034]ω4 isin [017 044] Attri-bute weights can be determined using Model (10)ω1 042ω2 007ω3 034ω4 017

Based on the attribute prospect value and attributeweights the comprehensive prospect value of each alter-native can be obtained using equation (11) whereV1 03994 V2 04211 and V3 03816 e ranking ofthree alternatives is V2 gtV1 gtV3 erefore V2 is the op-timal alternative

5 Comparison of Methods andSensitivity Analysis

51ComparisonofMethods To verify the effectiveness of theproposed approach we introduce two existing decision-making methods e first method is the double-referencepoint decision-making method based on the prospect theory[42] e second method is the TOPSIS method [43] In thetwo mentioned methods positive and negative ideal pointsare set as reference points used as the basis of the alternativeevaluation e positive and negative reference points ofattribute j are defined as 1113957rlowast+j and 1113957rlowastminusj respectively Based onthe prospect theory and TOPSIS method the comparisonbetween the attribute value of the alternative and the cor-responding reference points is analyzed and the evaluationvalue of each alternative is obtained In the double-referencepoint decision-making method based on the prospect

theory the maximum and minimum values of public ex-pectation are taken as positive and negative reference points

Public expectations obey the normal distributiontherefore the maximum and minimum values of publicexpectations can be obtained using the three-sigma (3σ)

theorem of normal distribution [44] For attribute j themaximum and minimum values of public expectations areobtained using the 3σ theorem of normal distributiondenoted by μ(rk

j) + 3σ(rkj) and μ(rk

j) minus 3σ(rkj) respectively

According to Definitions 1 and 2 the conversion values oflinguistic variables or uncertain linguistic variables are in therange [0 1] In the conversion crisp valuesrsquo distribution oflinguistic variables or uncertain linguistic variables in orderto ensure the maximum and minimum values we meet therange requirements in this type of distribution we define themaximum and minimum values as follows ifμ(rk

j ) + 3σ(rkj )gt 1 we set maximum value as 1 if

μ(rkj ) minus 3σ(rk

j )lt 0 we set minimum value as 0 For the otherscenarios the maximum and minimum values are set asμ(rk

j ) + 3σ(rkj ) and μ(rk

j) minus 3σ(rkj ) To satisfy the calculation

need of prospect value we set the positive and negativereference points of the attribute as interval numbers withequal upper and lower limits expressed as 1113957rlowast+j [(μ(rk

j) +

3σ(rkj)) (μ(rk

j) + 3σ(rkj))] and 1113957rlowastminusj [(μ(rk

j ) minus 3σ(rkj))

(μ(rkj ) minus 3σ(rk

j ))] In the TOPSISmethod we set the positiveand negative ideal points according to the maximum andminimum values of each attribute in all alternatives Forexample the positive and negative reference points of at-tribute j are respectively expressed as1113957rlowast+j [maxi12MyiU

j maxi12MyiUj ] and

1113957rlowastminusj [mini12MyiLj mini12MyiL

j ] e alternativeranking results of different decision methods are listed inTable 9

According to Table 9 the alternative ranking resultscorresponding to the existing methods proposed are

Table 8 Normalized attribute reference points and attribute values for different alternatives

AlternativeAverage distance between the stationand the population gathering area

along the line (m)

Estimated constructiontime of the project

(months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

1 [10000 10000] [00458 07137] [00318 02734] [03243 10000]2 [06182 06182] [06183 10000] [04943 10000] [10000 10000]3 [09532 09532] [03321 07137] [00318 04179] [03243 10000]Attributereference points [00000 01922] [00000 03798] [00000 04353] [00000 06486]

Table 7 Results of public expectation reference points for different attributes

AttributesAverage distance between the stationand the population gathering area

along the line (m)

Estimatedconstruction time ofthe project (months)

Construction-enclosedpublic areas (m2)

Dust and sandtreatmenteffect

(qualitative index)

Statistical distribution Normal distribution N (4460 74412) Normal distributionN (6949 8482)

Normal distribution N(28990 28992)

Normal distributionN (075 0152)

Comprehensive ambiguityof the expectations 1234 199 100621 006

Attribute reference point [43367 45834] [6750 7148] [2798379 2999621] [069 081]

10 Mathematical Problems in Engineering

inconsistent with the results obtained in this studyemainreason for the inconsistency of decision results is that thereference point setting in each decision method is different

e setting principle of reference points for the twoexisting methods is that reference points are obtainedaccording to the public expectation or the attribute value of thealternative e setting principle for the reference points usedin the two existing methods is relatively simple but it does notconsider the characteristics of public expectations For deci-sion-making activities in major public affairs public expec-tations must be fully considered to ensure the effectiveness ofthe decision-making results erefore public expectationsmust be considered when setting reference points

Based on the above analysis we develop the settingprinciple for reference points by considering public ex-pectations First the information on public expectations isdiscrete and has the characteristics of a normal distributionthus the mean value of public expectations is taken as thestandard setting for the reference point so that the referencepoint can effectively reflect the expectations of the publicgroup Second due to the uncertainty of the expectationevaluation by public individuals the expected value isusually expressed in an interval number which reflects theambiguity of public expectations e comprehensive ex-pected ambiguity of the public is used to represent theuncertain characteristics of public expectations e refer-ence point is set by the mean value of public expectationsand the comprehensive expected ambiguity of the public tobetter reflect the actual situation of public expectation ex-pression e results of the reference points are presented asinterval numbers e value of the reference points reflectsthe expectations of the public group and considers theuncertainty of the public evaluation

If an attribute value in interval form falls within therange of the corresponding reference points in whole or inpart we consider that part of the attribute value that fallswithin the range of the reference points exactly meetspublic expectations which means that the prospect valuefor this part of the attribute is zero erefore when theprospect value is calculated based on the interval reference

point the inclusion or cross relationship between theattribute interval and corresponding reference point in-terval should be considered For example Table 8 indicatesthat the normalized values of attribute 3 for alternatives 1and 3 are p13 [00318 02734] and p33 [00318

04179] respectively Moreover p13 and p33 are not equaland the normalized reference point for attribute 3 isq3 [00000 04353] It is apparent that p13 and p33 areentirely within the interval range of q3 We can affirm thatthe prospect values of attribute 3 for alternatives 1 and 3are both zero (v13 v33 0) In addition we use equations(7) and (8) and Table 8 to calculate the prospect values ofp13 and p33 we can also obtain the same result If wechoose the double-reference point decision-makingmethod to calculate the prospect value of p13 and p33 theresults of the prospect value for p13 and p33 arev13 minus 07329 and v33 minus 06732 which is different fromthe result of the proposed method erefore a differentsetting principle for the reference point leads to differentdecision-making results explaining why the sorting resultsare different in Table 9 erefore if we need to set ref-erence points for decision-making we must set themaccording to the actual situation for the decision problemand decision requirements to guarantee the effectiveness ofthe decision results

52 Sensitivity Analysis of the Reference Point Interval RangeAccording to the content above the comprehensive ambi-guity of public expectations is related to the interval range ofattribute reference points affecting the decision-makingresultse interval adjustment coefficient of reference pointθ (0le θle 1) is introduced in the expression of referencepoints to study the relationship between the value range ofthe reference points and decision results further e ref-erence point of attribute j is defined as1113957rlowastj [1113957rlowastLj 1113957rlowastUj ] [μ(rk

j ) minus θcj μ(rkj) + θcj] e compre-

hensive ambiguity of public expectations for each attribute cj

is known When the values of θ are 0 025 05 075 and 1the corresponding value range of reference points changes

Table 9 Alternative ranking results of different decision methods

Methods Attributereference points Decision results Ranking of

the alternatives

Double-reference point decision-makingmethod based on prospect theory

1113957rlowast+1 [22277 22277] 1113957r

lowast+2 [4405 4405]

1113957rlowast+3 [2029300 2029300] 1113957r

lowast+4 [100 100] V1 minus 01738

V2 minus 00983

V3 minus 01654

V2 gtV3 gtV11113957rlowastminus1 [66923 66923] 1113957r

lowastminus2 [9493 9493]

1113957rlowastminus3 [3768700 3768700] 1113957r

lowastminus4 [000 000]

TOPSIS

1113957rlowast+1 [33000 33000] 1113957r

lowast+2 [6100 6100]

1113957rlowast+3 [2537300 2537300] 1113957r

lowast+4 [0875 0875] V1 05719

V2 04772

V3 05564

V1 gtV3 gtV21113957rlowastminus1 [37900 37900] 1113957r

lowastminus2 [7100 7100]

1113957rlowastminus3 [2984900 2984900] 1113957r

lowastminus4 [075 075]

e proposed method 1113957rlowast1 [43367 45834] 1113957r

lowast2 [6750 7148]

1113957rlowast3 [2798379 2999621] 1113957r

lowast4 [069 081]

V1 03994

V2 04211

V3 03816

V2 gtV1 gtV3

Mathematical Problems in Engineering 11

accordingly e decision results for different conditions ofreference points are presented in Table 10

As listed in Table 10 when the interval adjustmentcoefficient of the reference points increases gradually theinterval range of the reference points for different attributesis also constantly enlarged so the alternative ranking resultsare not the same As observed in Table 10 if θ isin [0 05] thesorting result is V2 gtV3 gtV1 Moreover if θ isin [075 1] thesorting result is V2 gtV1 gtV3 In addition the alternativeranking results in Table 10 with the continuous expansion ofthe interval for reference points indicate that the compre-hensive prospect value of each alternative also constantlydecreases In addition the differentiation degree of thecomprehensive prospect value between alternatives alsogradually decreases erefore to ensure the effectiveness ofsetting attribute reference points we must consider theinfluence of the interval range of the reference points for thedecision results to reduce the difficulty of decision-makingand improve the accuracy of decision results

6 Conclusion

is paper proposed a new decision-making approach for amixed multiattribute decision-making problem with un-known attribute weights e advantages of this approachare summarized as follows First the reference point for eachattribute is set based on the distribution and comprehensiveambiguity of public expectations making the attributereference points better reflect the publicrsquos expected groupopinions and expectation uncertainty e effectiveness ofthe decision result is guaranteed Second in solving theattribute weights the attribute importance given by thepublic is used to determine the value range of the attributeweights so that the weighting results are in accordance withpublic opinions making the results of the attribute weightmore acceptable en the exact values of the attributeweights are determined to maximize the attribute infor-mation deviation of all the alternatives improving thediscrimination of alternatives ird each alternative is

evaluated based on the prospect theory to satisfy the cal-culation needs of the prospect valuee different expressionforms of the attribute information and reference point areunified and normalization is performed e operationabove can eliminate the influence of the expression form anddimension on decision-making making the decision op-erations smoother

e proposed approach also has certain limitationsFirst in the actual decision-making process the decision-making scenarios in some decision-making problems arenot static Changes in the decision-making scenarios easilycause decision risk which negatively affects the decisionactivity Our proposed approach does not consider dy-namic decision scenarios so the proposed approach mustbe further expanded and improved making it suitable forpublic-participation decision-making problems underchanging scenarios Second according to the settingprinciple of reference points the interval range of the at-tribute reference points affects the decision-making resultse interval range of the attribute reference points is largeand the discrimination of the evaluation results for dif-ferent alternatives is less obvious Because the intervalrange of attribute reference points is set by the compre-hensive ambiguity of public expectations if the dispersiondegree of public expectations is overly high (ie thecomprehensive ambiguity of public expectations is overlyhigh) the interval range of attribute reference points is alsoset over wide corresponding values is outcome may leadto a lack of differentiation in the decision results ereforeto ensure the effectiveness of the decision-making resultsreasonable interval ranges must be set for attribute refer-ence points according to the characteristics of public ex-pectation information which is also worth an in-depthstudy

Data Availability

e data used to support the findings of this study are in-cluded within the manuscript

Table 10 Decision results for different conditions of reference points

θ Reference points Comprehensive prospect value Alternatives ranking

0 1113957rlowast1 [44600 44600] 1113957r

lowast2 [6949 6949]

1113957rlowast3 [2899000 2899000] 1113957r

lowast4 [075 075]

V1 04365

V2 06769

V3 04699V2 gtV3 gtV1

025 1113957rlowast1 [44292 44909] 1113957r

lowast2 [6900 7000]

1113957rlowast3 [2873845 2924155] 1113957r

lowast4 [074 077]

V1 04138

V2 06068

V3 04442V2 gtV3 gtV1

050 1113957rlowast1 [43983 45217] 1113957r

lowast2 [6851 7050]

1113957rlowast3 [2848690 2949311] 1113957r

lowast4 [072 078]

V1 04104

V2 05425

V3 04231V2 gtV3 gtV1

075 1113957rlowast1 [43675 45526] 1113957r

lowast2 [6801 7099]

1113957rlowast3 [2823534 2974466] 1113957r

lowast4 [071 080]

V1 04141

V2 04827

V3 04066V2 gtV1 gtV3

1 1113957rlowast1 [43367 45834] 1113957r

lowast2 [6750 7148]

1113957rlowast3 [2798379 2999621] 1113957r

lowast4 [069 081]

V1 03994

V2 04211

V3 03816V2 gtV1 gtV3

12 Mathematical Problems in Engineering

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is research was funded by the National Natural ScienceFoundation of China (no 71902058) Natural ScienceFoundation of Hunan Province (nos 2018JJ3617 and2021JJ41088) Social Science Foundation of Hunan Province(no 19YBQ113) Scientific Research Foundation of HunanEducation Department (nos 18B484 19A06 and 20B155)and Social Science Achievements Appraisal CommitteeFoundation of Hunan Provincial Department (noXSP21YBC218)

References

[1] S Song Z Guo and X Wang ldquoRetracted article the cor-relation between social transformation economic risk andinternet public opinionrdquo Behaviour amp Information Technol-ogy vol 40 no 7 pp 723ndash733 2020

[2] W Chen F Tu and P Zheng ldquoA transnational networkedpublic sphere of air pollution analysis of a Twitter network ofPM25 from the risk society perspectiverdquo InformationCommunication amp Society vol 20 no 7 pp 1005ndash1023 2017

[3] X Yao J He and C Bao ldquoPublic participation modes inChinarsquos environmental impact assessment process an ana-lytical framework based on participation extent and conflictlevelrdquo Environmental Impact Assessment Review vol 84p 106400 2020

[4] X Zhang J G Xu and Y Ju ldquoPublic participation in NIMBYrisk mitigation A discourse zoning approach in the Chinesecontextrdquo Land Use Policy vol 77 pp 559ndash575 2018

[5] T Webler and S Tuler ldquoFour decades of public participationin risk decision makingrdquo Risk Analysis vol 41 no 3pp 503ndash518 2021

[6] J P Voszlig and N Amelung ldquoInnovating public participationmethods Technoscientization and reflexive engagementrdquoSocial Studies of Science vol 46 no 5 pp 749ndash772 2016

[7] A M Rıos B Benito and F Bastida ldquoFactors explainingpublic participation in the central government budget pro-cessrdquo Australian Journal of Public Administration vol 76no 1 pp 48ndash64 2017

[8] Y Zhou L Hou Y Yang H-Y Chong and S Moon ldquoAcomparative review and framework development on publicparticipation for decision-making in Chinese public projectsrdquoEnvironmental Impact Assessment Review vol 75 pp 79ndash872019

[9] T H You J Zhang and Z P Fan ldquoMulti-attribute onlinereview decision making method based on sentiment analysisand evidence theoryrdquo Journal of Systems amp Managementvol 28 no 3 pp 536ndash544 2019 in Chinese

[10] R L Charney T Rebmann P Dalawari and A EndrizalldquoPublic expectations of hospitals to provide resources andservices to the uninjured during disasters A qualitativestudyrdquo Health Security vol 14 no 6 pp 389ndash396 2016

[11] A Yildiz E Ayyildiz A Taskin Gumus and C Ozkan ldquoAframework to prioritize the public expectations from watertreatment plants based on trapezoidal type-2 fuzzy ahpmethodrdquo Environmental Management vol 67 no 3pp 439ndash448 2020

[12] Z Zhang Y Gao and Z L Li ldquoConsensus reaching for socialnetwork group decision making by considering leadershipand bounded confidencerdquo Knowledge-Based Systems vol 204pp 1ndash12 2020

[13] J J Zhu Z Z Ma H H Wang and Y Chen ldquoRisk decision-making method using interval numbers and its applicationbased on the prospect value with multiple reference pointsrdquoInformation Sciences vol 12 no 3 pp 385-386 2017

[14] X Li and X Chen ldquoValue determination method based onmultiple reference points under a trapezoidal intuitionisticfuzzy environmentrdquo Applied Soft Computing vol 63pp 39ndash49 2018

[15] J Gao Z Xu and H Liao ldquoA dynamic reference pointmethod for emergency response under hesitant probabilisticfuzzy environmentrdquo International Journal of Fuzzy Systemsvol 19 no 5 pp 1261ndash1278 2017

[16] C H Li W Li M J Li et al ldquoTarget-oriented model andapproach for attribute value evaluation with multiple refer-ence pointsrdquo Chinese Journal of Management Science vol 25no 7 pp 163ndash175 2017 in Chinese

[17] E Mastrocinque F J Ramırez A Honrubia-Escribano andD T Pham ldquoAn AHP-based multi-criteria model for sus-tainable supply chain development in the renewable energysectorrdquo Expert Systems with Applications vol 150 pp 1ndash172020

[18] L Chen Z Li and X Deng ldquoEmergency alternative evalu-ation under group decision makers a new method based onentropy weight and dematelrdquo International Journal of SystemsScience vol 51 no 3 pp 570ndash583 2020

[19] X KWang Y TWang J QWang P F Cheng and L Li ldquoATODIM-PROMETHEE II based multi-criteria group decisionmaking method for risk evaluation of water resource carryingcapacity under probabilistic linguistic Z-number circum-stancesrdquo Mathematics vol 8 no 7 p 1190 2020

[20] Y Liu Y Dong H Liang F Chiclana and E Herrera-Viedma ldquoMultiple attribute strategic weight manipulationwith minimum cost in a group decision making context withinterval attribute weights informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 49 no 10pp 1981ndash1992 2018

[21] F Meng C Tan and X Chen ldquoAn approach to Atanassovrsquosinterval-valued intuitionistic fuzzy multi-attribute decisionmaking based on prospect theoryrdquo International Journal ofComputational Intelligence Systems vol 8 no 3 pp 591ndash6052015

[22] N Zarbakhshnia Y Wu K Govindan and H Soleimani ldquoAnovel hybrid multiple attribute decision-making approach foroutsourcing sustainable reverse logisticsrdquo Journal of CleanerProduction vol 242 Article ID 118461 2020

[23] J J H Liou Y C Chuang E K Zavadskas and G H TzengldquoData-driven hybrid multiple attribute decision-makingmodel for green supplier evaluation and performance im-provementrdquo Journal of Cleaner Production vol 241 ArticleID 118321 2019

[24] Y H Pan and X L Geng ldquoHybrid multiple attribute decisionmaking approach based on Mo-RVIKORrdquo Chinese Journal ofManagement Science vol 27 no 12 pp 143ndash151 2019

[25] M Akram and D Shumaiza ldquoMulti-criteria decision makingbased on q-rung orthopair fuzzy promethee approachrdquo Ira-nian Journal of Fuzzy Systems vol 18 no 5 pp 107ndash127 2021

[26] C Jana G Muhiuddin and M Pal ldquoMulti-criteria decisionmaking approach based on SVTrN Dombi aggregationfunctionsrdquo Artificial Intelligence Review vol 54 no 5pp 3685ndash3723 2021

Mathematical Problems in Engineering 13

[27] C Jana G Muhiuddin and M Pal ldquoMultiple-attribute de-cision making problems based on SVTNH methodsrdquo Journalof Ambient Intelligence and Humanized Computing vol 11no 9 pp 3717ndash3733 2020

[28] C Jana GMuhiuddin andM Pal ldquoSomeDombi aggregationof Q -rung orthopair fuzzy numbers in multiple-attributedecision makingrdquo International Journal of Intelligent Systemsvol 34 no 12 pp 3220ndash3240 2019

[29] G Shahzadi G Muhiuddin M Arif Butt and A AshrafldquoHamacher interactive hybrid weighted averaging operatorsunder fermatean fuzzy numbersrdquo Journal of Mathematicsvol 2021 no 10 17 pages Article ID 5556017 2021

[30] K Zhang J Zhan and X Wang ldquoTOPSIS-WAA methodbased on a covering-based fuzzy rough set An application torating problemrdquo Information Sciences vol 539 pp 397ndash4212020

[31] S F Huang ldquoUsing Linguistic VIKOR and fuzzy cognitivemaps to select virtual reality games development projectrdquoMathematics vol 9 no 11 p 1253 2021

[32] M Akram A Luqman and C Kahraman ldquoHesitant py-thagorean fuzzy ELECTRE-II method for multi-criteria de-cision-making problemsrdquo Applied Soft Computing vol 108Article ID 107479 2021

[33] C Erdin and G Ozkaya ldquoTurkeyrsquos 2023 energy strategies andinvestment opportunities for renewable energy sources Siteselection based on ELECTRErdquo Sustainability vol 11 no 7Article ID 2136 2019

[34] T Wang H Li X Zhou D Liu and B Huang ldquoree-waydecision based on third-generation prospect theory with Z-numbersrdquo Information Sciences vol 569 pp 13ndash38 2021

[35] Z J Du S M Yu H Y Luo and X D Lin ldquoConsensusconvergence in large-group social network environmentcoordination between trust relationship and opinion simi-larityrdquo Knowledge-Based Systems vol 217 Article ID 1068282021

[36] J Wu F Chiclana H Fujita and E Herrera-Viedma ldquoAvisual interaction consensus model for social network groupdecision making with trust propagationrdquo Knowledge-BasedSystems vol 122 pp 39ndash50 2017

[37] L X Chen and N F Luo ldquoPythagorean fuzzy multi-criteriadecision-making based on prospect theoryrdquo Systems Engi-neering Keory amp Practice vol 40 no 3 pp 726ndash735 2020 inChinese

[38] Z S Xu Linguistic Decision Making Springer-Verlag BerlinGermany 2012

[39] R E Moore Methods and Applications of Interval AnalysisPrentice-Hall Hoboken NY USA 1979

[40] R R Ren W W Li M Zhao and X Li ldquoA large groupdecision making method based on public evaluationrdquoManagement Review vol 30 no 10 pp 238ndash247 2018

[41] L Wang Y M Wang and L Martınez ldquoA group decisionmethod based on prospect theory for emergency situationsrdquoInformation Sciences vol 418-419 pp 119ndash135 2017

[42] Y Wu C Xu and T Zhang ldquoEvaluation of renewable powersources using a fuzzy MCDM based on cumulative prospecttheory a case in Chinardquo Energy vol 147 pp 1227ndash1239 2018

[43] M M Salih B B Zaidan A A Zaidan and M A AhmedldquoSurvey on fuzzy TOPSIS state-of-the-art between 2007 and2017rdquo Computers amp Operations Research vol 104 pp 207ndash227 2019

[44] H Xiao Y Zhang X Liu H Yin P Liu and D C Liu ldquoArapid ultrasound vascular disease screening method usingPauTa Criterionrdquo Journal of Physics Conference Seriesvol 1769 Article ID 012009 2021

14 Mathematical Problems in Engineering

Table 6 e public expected values for different attributes in the form of crisp numbers

Average distance betweenthe station and the

population gathering areaalong the line (m)

Estimated construction timeof the project (months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

No Expected value No Expected value No Expected value No Expected value1 476 1 56 1 306585 1 056252 377 2 65 2 27748 2 05000

5471 349 6069 705 5517 319475 5490 068755472 356 6070 74 5518 29730 5491 07500

250 300 350 400 450 500 550 600 6500

20

40

60

80

100

120

140

160

180

Freq

uenc

y

Frequency

Public expectation of attribute 1 (m)

Normal distribution fitting curve

Figure 1 Fitting curve for public expected values of attribute 1

55 60 65 70 75 80 85 900

50

100

150

200

250

300

350

Freq

uenc

y

FrequencyNormal distribution fitting curve

Public expectation of attribute 2 (months)

Figure 2 Fitting curve for public expected values of attribute 2

22 24 26 28 3 32 34 36 380

20

40

60

80

100

120

140

Freq

uenc

y

Frequency

times104

Normal distribution fitting curve

Public expectation of attribute 3 (m2)

Figure 3 Fitting curve for public expected values of attribute 3

05 06 07 08 09 1 11 12Public expectation of attribute 4 (qualitative index)

0

500

1000

1500

2000

2500

Freq

uenc

y

FrequencyNormal distribution fitting curve

Figure 4 Fitting curve for public expected values of attribute 4

Mathematical Problems in Engineering 9

VM

08287 02070 00000 02164

04719 04752 03587 03983

07864 02070 00000 02164

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (12)

e value range of the attribute weights can be deter-mined using equation (9) based on the conversion betweenlinguistic variables and interval numbers ω1 isin [024 055]

ω2 isin [007 026] ω3 isin [010 034]ω4 isin [017 044] Attri-bute weights can be determined using Model (10)ω1 042ω2 007ω3 034ω4 017

Based on the attribute prospect value and attributeweights the comprehensive prospect value of each alter-native can be obtained using equation (11) whereV1 03994 V2 04211 and V3 03816 e ranking ofthree alternatives is V2 gtV1 gtV3 erefore V2 is the op-timal alternative

5 Comparison of Methods andSensitivity Analysis

51ComparisonofMethods To verify the effectiveness of theproposed approach we introduce two existing decision-making methods e first method is the double-referencepoint decision-making method based on the prospect theory[42] e second method is the TOPSIS method [43] In thetwo mentioned methods positive and negative ideal pointsare set as reference points used as the basis of the alternativeevaluation e positive and negative reference points ofattribute j are defined as 1113957rlowast+j and 1113957rlowastminusj respectively Based onthe prospect theory and TOPSIS method the comparisonbetween the attribute value of the alternative and the cor-responding reference points is analyzed and the evaluationvalue of each alternative is obtained In the double-referencepoint decision-making method based on the prospect

theory the maximum and minimum values of public ex-pectation are taken as positive and negative reference points

Public expectations obey the normal distributiontherefore the maximum and minimum values of publicexpectations can be obtained using the three-sigma (3σ)

theorem of normal distribution [44] For attribute j themaximum and minimum values of public expectations areobtained using the 3σ theorem of normal distributiondenoted by μ(rk

j) + 3σ(rkj) and μ(rk

j) minus 3σ(rkj) respectively

According to Definitions 1 and 2 the conversion values oflinguistic variables or uncertain linguistic variables are in therange [0 1] In the conversion crisp valuesrsquo distribution oflinguistic variables or uncertain linguistic variables in orderto ensure the maximum and minimum values we meet therange requirements in this type of distribution we define themaximum and minimum values as follows ifμ(rk

j ) + 3σ(rkj )gt 1 we set maximum value as 1 if

μ(rkj ) minus 3σ(rk

j )lt 0 we set minimum value as 0 For the otherscenarios the maximum and minimum values are set asμ(rk

j ) + 3σ(rkj ) and μ(rk

j) minus 3σ(rkj ) To satisfy the calculation

need of prospect value we set the positive and negativereference points of the attribute as interval numbers withequal upper and lower limits expressed as 1113957rlowast+j [(μ(rk

j) +

3σ(rkj)) (μ(rk

j) + 3σ(rkj))] and 1113957rlowastminusj [(μ(rk

j ) minus 3σ(rkj))

(μ(rkj ) minus 3σ(rk

j ))] In the TOPSISmethod we set the positiveand negative ideal points according to the maximum andminimum values of each attribute in all alternatives Forexample the positive and negative reference points of at-tribute j are respectively expressed as1113957rlowast+j [maxi12MyiU

j maxi12MyiUj ] and

1113957rlowastminusj [mini12MyiLj mini12MyiL

j ] e alternativeranking results of different decision methods are listed inTable 9

According to Table 9 the alternative ranking resultscorresponding to the existing methods proposed are

Table 8 Normalized attribute reference points and attribute values for different alternatives

AlternativeAverage distance between the stationand the population gathering area

along the line (m)

Estimated constructiontime of the project

(months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

1 [10000 10000] [00458 07137] [00318 02734] [03243 10000]2 [06182 06182] [06183 10000] [04943 10000] [10000 10000]3 [09532 09532] [03321 07137] [00318 04179] [03243 10000]Attributereference points [00000 01922] [00000 03798] [00000 04353] [00000 06486]

Table 7 Results of public expectation reference points for different attributes

AttributesAverage distance between the stationand the population gathering area

along the line (m)

Estimatedconstruction time ofthe project (months)

Construction-enclosedpublic areas (m2)

Dust and sandtreatmenteffect

(qualitative index)

Statistical distribution Normal distribution N (4460 74412) Normal distributionN (6949 8482)

Normal distribution N(28990 28992)

Normal distributionN (075 0152)

Comprehensive ambiguityof the expectations 1234 199 100621 006

Attribute reference point [43367 45834] [6750 7148] [2798379 2999621] [069 081]

10 Mathematical Problems in Engineering

inconsistent with the results obtained in this studyemainreason for the inconsistency of decision results is that thereference point setting in each decision method is different

e setting principle of reference points for the twoexisting methods is that reference points are obtainedaccording to the public expectation or the attribute value of thealternative e setting principle for the reference points usedin the two existing methods is relatively simple but it does notconsider the characteristics of public expectations For deci-sion-making activities in major public affairs public expec-tations must be fully considered to ensure the effectiveness ofthe decision-making results erefore public expectationsmust be considered when setting reference points

Based on the above analysis we develop the settingprinciple for reference points by considering public ex-pectations First the information on public expectations isdiscrete and has the characteristics of a normal distributionthus the mean value of public expectations is taken as thestandard setting for the reference point so that the referencepoint can effectively reflect the expectations of the publicgroup Second due to the uncertainty of the expectationevaluation by public individuals the expected value isusually expressed in an interval number which reflects theambiguity of public expectations e comprehensive ex-pected ambiguity of the public is used to represent theuncertain characteristics of public expectations e refer-ence point is set by the mean value of public expectationsand the comprehensive expected ambiguity of the public tobetter reflect the actual situation of public expectation ex-pression e results of the reference points are presented asinterval numbers e value of the reference points reflectsthe expectations of the public group and considers theuncertainty of the public evaluation

If an attribute value in interval form falls within therange of the corresponding reference points in whole or inpart we consider that part of the attribute value that fallswithin the range of the reference points exactly meetspublic expectations which means that the prospect valuefor this part of the attribute is zero erefore when theprospect value is calculated based on the interval reference

point the inclusion or cross relationship between theattribute interval and corresponding reference point in-terval should be considered For example Table 8 indicatesthat the normalized values of attribute 3 for alternatives 1and 3 are p13 [00318 02734] and p33 [00318

04179] respectively Moreover p13 and p33 are not equaland the normalized reference point for attribute 3 isq3 [00000 04353] It is apparent that p13 and p33 areentirely within the interval range of q3 We can affirm thatthe prospect values of attribute 3 for alternatives 1 and 3are both zero (v13 v33 0) In addition we use equations(7) and (8) and Table 8 to calculate the prospect values ofp13 and p33 we can also obtain the same result If wechoose the double-reference point decision-makingmethod to calculate the prospect value of p13 and p33 theresults of the prospect value for p13 and p33 arev13 minus 07329 and v33 minus 06732 which is different fromthe result of the proposed method erefore a differentsetting principle for the reference point leads to differentdecision-making results explaining why the sorting resultsare different in Table 9 erefore if we need to set ref-erence points for decision-making we must set themaccording to the actual situation for the decision problemand decision requirements to guarantee the effectiveness ofthe decision results

52 Sensitivity Analysis of the Reference Point Interval RangeAccording to the content above the comprehensive ambi-guity of public expectations is related to the interval range ofattribute reference points affecting the decision-makingresultse interval adjustment coefficient of reference pointθ (0le θle 1) is introduced in the expression of referencepoints to study the relationship between the value range ofthe reference points and decision results further e ref-erence point of attribute j is defined as1113957rlowastj [1113957rlowastLj 1113957rlowastUj ] [μ(rk

j ) minus θcj μ(rkj) + θcj] e compre-

hensive ambiguity of public expectations for each attribute cj

is known When the values of θ are 0 025 05 075 and 1the corresponding value range of reference points changes

Table 9 Alternative ranking results of different decision methods

Methods Attributereference points Decision results Ranking of

the alternatives

Double-reference point decision-makingmethod based on prospect theory

1113957rlowast+1 [22277 22277] 1113957r

lowast+2 [4405 4405]

1113957rlowast+3 [2029300 2029300] 1113957r

lowast+4 [100 100] V1 minus 01738

V2 minus 00983

V3 minus 01654

V2 gtV3 gtV11113957rlowastminus1 [66923 66923] 1113957r

lowastminus2 [9493 9493]

1113957rlowastminus3 [3768700 3768700] 1113957r

lowastminus4 [000 000]

TOPSIS

1113957rlowast+1 [33000 33000] 1113957r

lowast+2 [6100 6100]

1113957rlowast+3 [2537300 2537300] 1113957r

lowast+4 [0875 0875] V1 05719

V2 04772

V3 05564

V1 gtV3 gtV21113957rlowastminus1 [37900 37900] 1113957r

lowastminus2 [7100 7100]

1113957rlowastminus3 [2984900 2984900] 1113957r

lowastminus4 [075 075]

e proposed method 1113957rlowast1 [43367 45834] 1113957r

lowast2 [6750 7148]

1113957rlowast3 [2798379 2999621] 1113957r

lowast4 [069 081]

V1 03994

V2 04211

V3 03816

V2 gtV1 gtV3

Mathematical Problems in Engineering 11

accordingly e decision results for different conditions ofreference points are presented in Table 10

As listed in Table 10 when the interval adjustmentcoefficient of the reference points increases gradually theinterval range of the reference points for different attributesis also constantly enlarged so the alternative ranking resultsare not the same As observed in Table 10 if θ isin [0 05] thesorting result is V2 gtV3 gtV1 Moreover if θ isin [075 1] thesorting result is V2 gtV1 gtV3 In addition the alternativeranking results in Table 10 with the continuous expansion ofthe interval for reference points indicate that the compre-hensive prospect value of each alternative also constantlydecreases In addition the differentiation degree of thecomprehensive prospect value between alternatives alsogradually decreases erefore to ensure the effectiveness ofsetting attribute reference points we must consider theinfluence of the interval range of the reference points for thedecision results to reduce the difficulty of decision-makingand improve the accuracy of decision results

6 Conclusion

is paper proposed a new decision-making approach for amixed multiattribute decision-making problem with un-known attribute weights e advantages of this approachare summarized as follows First the reference point for eachattribute is set based on the distribution and comprehensiveambiguity of public expectations making the attributereference points better reflect the publicrsquos expected groupopinions and expectation uncertainty e effectiveness ofthe decision result is guaranteed Second in solving theattribute weights the attribute importance given by thepublic is used to determine the value range of the attributeweights so that the weighting results are in accordance withpublic opinions making the results of the attribute weightmore acceptable en the exact values of the attributeweights are determined to maximize the attribute infor-mation deviation of all the alternatives improving thediscrimination of alternatives ird each alternative is

evaluated based on the prospect theory to satisfy the cal-culation needs of the prospect valuee different expressionforms of the attribute information and reference point areunified and normalization is performed e operationabove can eliminate the influence of the expression form anddimension on decision-making making the decision op-erations smoother

e proposed approach also has certain limitationsFirst in the actual decision-making process the decision-making scenarios in some decision-making problems arenot static Changes in the decision-making scenarios easilycause decision risk which negatively affects the decisionactivity Our proposed approach does not consider dy-namic decision scenarios so the proposed approach mustbe further expanded and improved making it suitable forpublic-participation decision-making problems underchanging scenarios Second according to the settingprinciple of reference points the interval range of the at-tribute reference points affects the decision-making resultse interval range of the attribute reference points is largeand the discrimination of the evaluation results for dif-ferent alternatives is less obvious Because the intervalrange of attribute reference points is set by the compre-hensive ambiguity of public expectations if the dispersiondegree of public expectations is overly high (ie thecomprehensive ambiguity of public expectations is overlyhigh) the interval range of attribute reference points is alsoset over wide corresponding values is outcome may leadto a lack of differentiation in the decision results ereforeto ensure the effectiveness of the decision-making resultsreasonable interval ranges must be set for attribute refer-ence points according to the characteristics of public ex-pectation information which is also worth an in-depthstudy

Data Availability

e data used to support the findings of this study are in-cluded within the manuscript

Table 10 Decision results for different conditions of reference points

θ Reference points Comprehensive prospect value Alternatives ranking

0 1113957rlowast1 [44600 44600] 1113957r

lowast2 [6949 6949]

1113957rlowast3 [2899000 2899000] 1113957r

lowast4 [075 075]

V1 04365

V2 06769

V3 04699V2 gtV3 gtV1

025 1113957rlowast1 [44292 44909] 1113957r

lowast2 [6900 7000]

1113957rlowast3 [2873845 2924155] 1113957r

lowast4 [074 077]

V1 04138

V2 06068

V3 04442V2 gtV3 gtV1

050 1113957rlowast1 [43983 45217] 1113957r

lowast2 [6851 7050]

1113957rlowast3 [2848690 2949311] 1113957r

lowast4 [072 078]

V1 04104

V2 05425

V3 04231V2 gtV3 gtV1

075 1113957rlowast1 [43675 45526] 1113957r

lowast2 [6801 7099]

1113957rlowast3 [2823534 2974466] 1113957r

lowast4 [071 080]

V1 04141

V2 04827

V3 04066V2 gtV1 gtV3

1 1113957rlowast1 [43367 45834] 1113957r

lowast2 [6750 7148]

1113957rlowast3 [2798379 2999621] 1113957r

lowast4 [069 081]

V1 03994

V2 04211

V3 03816V2 gtV1 gtV3

12 Mathematical Problems in Engineering

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is research was funded by the National Natural ScienceFoundation of China (no 71902058) Natural ScienceFoundation of Hunan Province (nos 2018JJ3617 and2021JJ41088) Social Science Foundation of Hunan Province(no 19YBQ113) Scientific Research Foundation of HunanEducation Department (nos 18B484 19A06 and 20B155)and Social Science Achievements Appraisal CommitteeFoundation of Hunan Provincial Department (noXSP21YBC218)

References

[1] S Song Z Guo and X Wang ldquoRetracted article the cor-relation between social transformation economic risk andinternet public opinionrdquo Behaviour amp Information Technol-ogy vol 40 no 7 pp 723ndash733 2020

[2] W Chen F Tu and P Zheng ldquoA transnational networkedpublic sphere of air pollution analysis of a Twitter network ofPM25 from the risk society perspectiverdquo InformationCommunication amp Society vol 20 no 7 pp 1005ndash1023 2017

[3] X Yao J He and C Bao ldquoPublic participation modes inChinarsquos environmental impact assessment process an ana-lytical framework based on participation extent and conflictlevelrdquo Environmental Impact Assessment Review vol 84p 106400 2020

[4] X Zhang J G Xu and Y Ju ldquoPublic participation in NIMBYrisk mitigation A discourse zoning approach in the Chinesecontextrdquo Land Use Policy vol 77 pp 559ndash575 2018

[5] T Webler and S Tuler ldquoFour decades of public participationin risk decision makingrdquo Risk Analysis vol 41 no 3pp 503ndash518 2021

[6] J P Voszlig and N Amelung ldquoInnovating public participationmethods Technoscientization and reflexive engagementrdquoSocial Studies of Science vol 46 no 5 pp 749ndash772 2016

[7] A M Rıos B Benito and F Bastida ldquoFactors explainingpublic participation in the central government budget pro-cessrdquo Australian Journal of Public Administration vol 76no 1 pp 48ndash64 2017

[8] Y Zhou L Hou Y Yang H-Y Chong and S Moon ldquoAcomparative review and framework development on publicparticipation for decision-making in Chinese public projectsrdquoEnvironmental Impact Assessment Review vol 75 pp 79ndash872019

[9] T H You J Zhang and Z P Fan ldquoMulti-attribute onlinereview decision making method based on sentiment analysisand evidence theoryrdquo Journal of Systems amp Managementvol 28 no 3 pp 536ndash544 2019 in Chinese

[10] R L Charney T Rebmann P Dalawari and A EndrizalldquoPublic expectations of hospitals to provide resources andservices to the uninjured during disasters A qualitativestudyrdquo Health Security vol 14 no 6 pp 389ndash396 2016

[11] A Yildiz E Ayyildiz A Taskin Gumus and C Ozkan ldquoAframework to prioritize the public expectations from watertreatment plants based on trapezoidal type-2 fuzzy ahpmethodrdquo Environmental Management vol 67 no 3pp 439ndash448 2020

[12] Z Zhang Y Gao and Z L Li ldquoConsensus reaching for socialnetwork group decision making by considering leadershipand bounded confidencerdquo Knowledge-Based Systems vol 204pp 1ndash12 2020

[13] J J Zhu Z Z Ma H H Wang and Y Chen ldquoRisk decision-making method using interval numbers and its applicationbased on the prospect value with multiple reference pointsrdquoInformation Sciences vol 12 no 3 pp 385-386 2017

[14] X Li and X Chen ldquoValue determination method based onmultiple reference points under a trapezoidal intuitionisticfuzzy environmentrdquo Applied Soft Computing vol 63pp 39ndash49 2018

[15] J Gao Z Xu and H Liao ldquoA dynamic reference pointmethod for emergency response under hesitant probabilisticfuzzy environmentrdquo International Journal of Fuzzy Systemsvol 19 no 5 pp 1261ndash1278 2017

[16] C H Li W Li M J Li et al ldquoTarget-oriented model andapproach for attribute value evaluation with multiple refer-ence pointsrdquo Chinese Journal of Management Science vol 25no 7 pp 163ndash175 2017 in Chinese

[17] E Mastrocinque F J Ramırez A Honrubia-Escribano andD T Pham ldquoAn AHP-based multi-criteria model for sus-tainable supply chain development in the renewable energysectorrdquo Expert Systems with Applications vol 150 pp 1ndash172020

[18] L Chen Z Li and X Deng ldquoEmergency alternative evalu-ation under group decision makers a new method based onentropy weight and dematelrdquo International Journal of SystemsScience vol 51 no 3 pp 570ndash583 2020

[19] X KWang Y TWang J QWang P F Cheng and L Li ldquoATODIM-PROMETHEE II based multi-criteria group decisionmaking method for risk evaluation of water resource carryingcapacity under probabilistic linguistic Z-number circum-stancesrdquo Mathematics vol 8 no 7 p 1190 2020

[20] Y Liu Y Dong H Liang F Chiclana and E Herrera-Viedma ldquoMultiple attribute strategic weight manipulationwith minimum cost in a group decision making context withinterval attribute weights informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 49 no 10pp 1981ndash1992 2018

[21] F Meng C Tan and X Chen ldquoAn approach to Atanassovrsquosinterval-valued intuitionistic fuzzy multi-attribute decisionmaking based on prospect theoryrdquo International Journal ofComputational Intelligence Systems vol 8 no 3 pp 591ndash6052015

[22] N Zarbakhshnia Y Wu K Govindan and H Soleimani ldquoAnovel hybrid multiple attribute decision-making approach foroutsourcing sustainable reverse logisticsrdquo Journal of CleanerProduction vol 242 Article ID 118461 2020

[23] J J H Liou Y C Chuang E K Zavadskas and G H TzengldquoData-driven hybrid multiple attribute decision-makingmodel for green supplier evaluation and performance im-provementrdquo Journal of Cleaner Production vol 241 ArticleID 118321 2019

[24] Y H Pan and X L Geng ldquoHybrid multiple attribute decisionmaking approach based on Mo-RVIKORrdquo Chinese Journal ofManagement Science vol 27 no 12 pp 143ndash151 2019

[25] M Akram and D Shumaiza ldquoMulti-criteria decision makingbased on q-rung orthopair fuzzy promethee approachrdquo Ira-nian Journal of Fuzzy Systems vol 18 no 5 pp 107ndash127 2021

[26] C Jana G Muhiuddin and M Pal ldquoMulti-criteria decisionmaking approach based on SVTrN Dombi aggregationfunctionsrdquo Artificial Intelligence Review vol 54 no 5pp 3685ndash3723 2021

Mathematical Problems in Engineering 13

[27] C Jana G Muhiuddin and M Pal ldquoMultiple-attribute de-cision making problems based on SVTNH methodsrdquo Journalof Ambient Intelligence and Humanized Computing vol 11no 9 pp 3717ndash3733 2020

[28] C Jana GMuhiuddin andM Pal ldquoSomeDombi aggregationof Q -rung orthopair fuzzy numbers in multiple-attributedecision makingrdquo International Journal of Intelligent Systemsvol 34 no 12 pp 3220ndash3240 2019

[29] G Shahzadi G Muhiuddin M Arif Butt and A AshrafldquoHamacher interactive hybrid weighted averaging operatorsunder fermatean fuzzy numbersrdquo Journal of Mathematicsvol 2021 no 10 17 pages Article ID 5556017 2021

[30] K Zhang J Zhan and X Wang ldquoTOPSIS-WAA methodbased on a covering-based fuzzy rough set An application torating problemrdquo Information Sciences vol 539 pp 397ndash4212020

[31] S F Huang ldquoUsing Linguistic VIKOR and fuzzy cognitivemaps to select virtual reality games development projectrdquoMathematics vol 9 no 11 p 1253 2021

[32] M Akram A Luqman and C Kahraman ldquoHesitant py-thagorean fuzzy ELECTRE-II method for multi-criteria de-cision-making problemsrdquo Applied Soft Computing vol 108Article ID 107479 2021

[33] C Erdin and G Ozkaya ldquoTurkeyrsquos 2023 energy strategies andinvestment opportunities for renewable energy sources Siteselection based on ELECTRErdquo Sustainability vol 11 no 7Article ID 2136 2019

[34] T Wang H Li X Zhou D Liu and B Huang ldquoree-waydecision based on third-generation prospect theory with Z-numbersrdquo Information Sciences vol 569 pp 13ndash38 2021

[35] Z J Du S M Yu H Y Luo and X D Lin ldquoConsensusconvergence in large-group social network environmentcoordination between trust relationship and opinion simi-larityrdquo Knowledge-Based Systems vol 217 Article ID 1068282021

[36] J Wu F Chiclana H Fujita and E Herrera-Viedma ldquoAvisual interaction consensus model for social network groupdecision making with trust propagationrdquo Knowledge-BasedSystems vol 122 pp 39ndash50 2017

[37] L X Chen and N F Luo ldquoPythagorean fuzzy multi-criteriadecision-making based on prospect theoryrdquo Systems Engi-neering Keory amp Practice vol 40 no 3 pp 726ndash735 2020 inChinese

[38] Z S Xu Linguistic Decision Making Springer-Verlag BerlinGermany 2012

[39] R E Moore Methods and Applications of Interval AnalysisPrentice-Hall Hoboken NY USA 1979

[40] R R Ren W W Li M Zhao and X Li ldquoA large groupdecision making method based on public evaluationrdquoManagement Review vol 30 no 10 pp 238ndash247 2018

[41] L Wang Y M Wang and L Martınez ldquoA group decisionmethod based on prospect theory for emergency situationsrdquoInformation Sciences vol 418-419 pp 119ndash135 2017

[42] Y Wu C Xu and T Zhang ldquoEvaluation of renewable powersources using a fuzzy MCDM based on cumulative prospecttheory a case in Chinardquo Energy vol 147 pp 1227ndash1239 2018

[43] M M Salih B B Zaidan A A Zaidan and M A AhmedldquoSurvey on fuzzy TOPSIS state-of-the-art between 2007 and2017rdquo Computers amp Operations Research vol 104 pp 207ndash227 2019

[44] H Xiao Y Zhang X Liu H Yin P Liu and D C Liu ldquoArapid ultrasound vascular disease screening method usingPauTa Criterionrdquo Journal of Physics Conference Seriesvol 1769 Article ID 012009 2021

14 Mathematical Problems in Engineering

VM

08287 02070 00000 02164

04719 04752 03587 03983

07864 02070 00000 02164

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (12)

e value range of the attribute weights can be deter-mined using equation (9) based on the conversion betweenlinguistic variables and interval numbers ω1 isin [024 055]

ω2 isin [007 026] ω3 isin [010 034]ω4 isin [017 044] Attri-bute weights can be determined using Model (10)ω1 042ω2 007ω3 034ω4 017

Based on the attribute prospect value and attributeweights the comprehensive prospect value of each alter-native can be obtained using equation (11) whereV1 03994 V2 04211 and V3 03816 e ranking ofthree alternatives is V2 gtV1 gtV3 erefore V2 is the op-timal alternative

5 Comparison of Methods andSensitivity Analysis

51ComparisonofMethods To verify the effectiveness of theproposed approach we introduce two existing decision-making methods e first method is the double-referencepoint decision-making method based on the prospect theory[42] e second method is the TOPSIS method [43] In thetwo mentioned methods positive and negative ideal pointsare set as reference points used as the basis of the alternativeevaluation e positive and negative reference points ofattribute j are defined as 1113957rlowast+j and 1113957rlowastminusj respectively Based onthe prospect theory and TOPSIS method the comparisonbetween the attribute value of the alternative and the cor-responding reference points is analyzed and the evaluationvalue of each alternative is obtained In the double-referencepoint decision-making method based on the prospect

theory the maximum and minimum values of public ex-pectation are taken as positive and negative reference points

Public expectations obey the normal distributiontherefore the maximum and minimum values of publicexpectations can be obtained using the three-sigma (3σ)

theorem of normal distribution [44] For attribute j themaximum and minimum values of public expectations areobtained using the 3σ theorem of normal distributiondenoted by μ(rk

j) + 3σ(rkj) and μ(rk

j) minus 3σ(rkj) respectively

According to Definitions 1 and 2 the conversion values oflinguistic variables or uncertain linguistic variables are in therange [0 1] In the conversion crisp valuesrsquo distribution oflinguistic variables or uncertain linguistic variables in orderto ensure the maximum and minimum values we meet therange requirements in this type of distribution we define themaximum and minimum values as follows ifμ(rk

j ) + 3σ(rkj )gt 1 we set maximum value as 1 if

μ(rkj ) minus 3σ(rk

j )lt 0 we set minimum value as 0 For the otherscenarios the maximum and minimum values are set asμ(rk

j ) + 3σ(rkj ) and μ(rk

j) minus 3σ(rkj ) To satisfy the calculation

need of prospect value we set the positive and negativereference points of the attribute as interval numbers withequal upper and lower limits expressed as 1113957rlowast+j [(μ(rk

j) +

3σ(rkj)) (μ(rk

j) + 3σ(rkj))] and 1113957rlowastminusj [(μ(rk

j ) minus 3σ(rkj))

(μ(rkj ) minus 3σ(rk

j ))] In the TOPSISmethod we set the positiveand negative ideal points according to the maximum andminimum values of each attribute in all alternatives Forexample the positive and negative reference points of at-tribute j are respectively expressed as1113957rlowast+j [maxi12MyiU

j maxi12MyiUj ] and

1113957rlowastminusj [mini12MyiLj mini12MyiL

j ] e alternativeranking results of different decision methods are listed inTable 9

According to Table 9 the alternative ranking resultscorresponding to the existing methods proposed are

Table 8 Normalized attribute reference points and attribute values for different alternatives

AlternativeAverage distance between the stationand the population gathering area

along the line (m)

Estimated constructiontime of the project

(months)

Construction-enclosedpublic areas (m2)

Dust and sand treatmenteffect (qualitative index)

1 [10000 10000] [00458 07137] [00318 02734] [03243 10000]2 [06182 06182] [06183 10000] [04943 10000] [10000 10000]3 [09532 09532] [03321 07137] [00318 04179] [03243 10000]Attributereference points [00000 01922] [00000 03798] [00000 04353] [00000 06486]

Table 7 Results of public expectation reference points for different attributes

AttributesAverage distance between the stationand the population gathering area

along the line (m)

Estimatedconstruction time ofthe project (months)

Construction-enclosedpublic areas (m2)

Dust and sandtreatmenteffect

(qualitative index)

Statistical distribution Normal distribution N (4460 74412) Normal distributionN (6949 8482)

Normal distribution N(28990 28992)

Normal distributionN (075 0152)

Comprehensive ambiguityof the expectations 1234 199 100621 006

Attribute reference point [43367 45834] [6750 7148] [2798379 2999621] [069 081]

10 Mathematical Problems in Engineering

inconsistent with the results obtained in this studyemainreason for the inconsistency of decision results is that thereference point setting in each decision method is different

e setting principle of reference points for the twoexisting methods is that reference points are obtainedaccording to the public expectation or the attribute value of thealternative e setting principle for the reference points usedin the two existing methods is relatively simple but it does notconsider the characteristics of public expectations For deci-sion-making activities in major public affairs public expec-tations must be fully considered to ensure the effectiveness ofthe decision-making results erefore public expectationsmust be considered when setting reference points

Based on the above analysis we develop the settingprinciple for reference points by considering public ex-pectations First the information on public expectations isdiscrete and has the characteristics of a normal distributionthus the mean value of public expectations is taken as thestandard setting for the reference point so that the referencepoint can effectively reflect the expectations of the publicgroup Second due to the uncertainty of the expectationevaluation by public individuals the expected value isusually expressed in an interval number which reflects theambiguity of public expectations e comprehensive ex-pected ambiguity of the public is used to represent theuncertain characteristics of public expectations e refer-ence point is set by the mean value of public expectationsand the comprehensive expected ambiguity of the public tobetter reflect the actual situation of public expectation ex-pression e results of the reference points are presented asinterval numbers e value of the reference points reflectsthe expectations of the public group and considers theuncertainty of the public evaluation

If an attribute value in interval form falls within therange of the corresponding reference points in whole or inpart we consider that part of the attribute value that fallswithin the range of the reference points exactly meetspublic expectations which means that the prospect valuefor this part of the attribute is zero erefore when theprospect value is calculated based on the interval reference

point the inclusion or cross relationship between theattribute interval and corresponding reference point in-terval should be considered For example Table 8 indicatesthat the normalized values of attribute 3 for alternatives 1and 3 are p13 [00318 02734] and p33 [00318

04179] respectively Moreover p13 and p33 are not equaland the normalized reference point for attribute 3 isq3 [00000 04353] It is apparent that p13 and p33 areentirely within the interval range of q3 We can affirm thatthe prospect values of attribute 3 for alternatives 1 and 3are both zero (v13 v33 0) In addition we use equations(7) and (8) and Table 8 to calculate the prospect values ofp13 and p33 we can also obtain the same result If wechoose the double-reference point decision-makingmethod to calculate the prospect value of p13 and p33 theresults of the prospect value for p13 and p33 arev13 minus 07329 and v33 minus 06732 which is different fromthe result of the proposed method erefore a differentsetting principle for the reference point leads to differentdecision-making results explaining why the sorting resultsare different in Table 9 erefore if we need to set ref-erence points for decision-making we must set themaccording to the actual situation for the decision problemand decision requirements to guarantee the effectiveness ofthe decision results

52 Sensitivity Analysis of the Reference Point Interval RangeAccording to the content above the comprehensive ambi-guity of public expectations is related to the interval range ofattribute reference points affecting the decision-makingresultse interval adjustment coefficient of reference pointθ (0le θle 1) is introduced in the expression of referencepoints to study the relationship between the value range ofthe reference points and decision results further e ref-erence point of attribute j is defined as1113957rlowastj [1113957rlowastLj 1113957rlowastUj ] [μ(rk

j ) minus θcj μ(rkj) + θcj] e compre-

hensive ambiguity of public expectations for each attribute cj

is known When the values of θ are 0 025 05 075 and 1the corresponding value range of reference points changes

Table 9 Alternative ranking results of different decision methods

Methods Attributereference points Decision results Ranking of

the alternatives

Double-reference point decision-makingmethod based on prospect theory

1113957rlowast+1 [22277 22277] 1113957r

lowast+2 [4405 4405]

1113957rlowast+3 [2029300 2029300] 1113957r

lowast+4 [100 100] V1 minus 01738

V2 minus 00983

V3 minus 01654

V2 gtV3 gtV11113957rlowastminus1 [66923 66923] 1113957r

lowastminus2 [9493 9493]

1113957rlowastminus3 [3768700 3768700] 1113957r

lowastminus4 [000 000]

TOPSIS

1113957rlowast+1 [33000 33000] 1113957r

lowast+2 [6100 6100]

1113957rlowast+3 [2537300 2537300] 1113957r

lowast+4 [0875 0875] V1 05719

V2 04772

V3 05564

V1 gtV3 gtV21113957rlowastminus1 [37900 37900] 1113957r

lowastminus2 [7100 7100]

1113957rlowastminus3 [2984900 2984900] 1113957r

lowastminus4 [075 075]

e proposed method 1113957rlowast1 [43367 45834] 1113957r

lowast2 [6750 7148]

1113957rlowast3 [2798379 2999621] 1113957r

lowast4 [069 081]

V1 03994

V2 04211

V3 03816

V2 gtV1 gtV3

Mathematical Problems in Engineering 11

accordingly e decision results for different conditions ofreference points are presented in Table 10

As listed in Table 10 when the interval adjustmentcoefficient of the reference points increases gradually theinterval range of the reference points for different attributesis also constantly enlarged so the alternative ranking resultsare not the same As observed in Table 10 if θ isin [0 05] thesorting result is V2 gtV3 gtV1 Moreover if θ isin [075 1] thesorting result is V2 gtV1 gtV3 In addition the alternativeranking results in Table 10 with the continuous expansion ofthe interval for reference points indicate that the compre-hensive prospect value of each alternative also constantlydecreases In addition the differentiation degree of thecomprehensive prospect value between alternatives alsogradually decreases erefore to ensure the effectiveness ofsetting attribute reference points we must consider theinfluence of the interval range of the reference points for thedecision results to reduce the difficulty of decision-makingand improve the accuracy of decision results

6 Conclusion

is paper proposed a new decision-making approach for amixed multiattribute decision-making problem with un-known attribute weights e advantages of this approachare summarized as follows First the reference point for eachattribute is set based on the distribution and comprehensiveambiguity of public expectations making the attributereference points better reflect the publicrsquos expected groupopinions and expectation uncertainty e effectiveness ofthe decision result is guaranteed Second in solving theattribute weights the attribute importance given by thepublic is used to determine the value range of the attributeweights so that the weighting results are in accordance withpublic opinions making the results of the attribute weightmore acceptable en the exact values of the attributeweights are determined to maximize the attribute infor-mation deviation of all the alternatives improving thediscrimination of alternatives ird each alternative is

evaluated based on the prospect theory to satisfy the cal-culation needs of the prospect valuee different expressionforms of the attribute information and reference point areunified and normalization is performed e operationabove can eliminate the influence of the expression form anddimension on decision-making making the decision op-erations smoother

e proposed approach also has certain limitationsFirst in the actual decision-making process the decision-making scenarios in some decision-making problems arenot static Changes in the decision-making scenarios easilycause decision risk which negatively affects the decisionactivity Our proposed approach does not consider dy-namic decision scenarios so the proposed approach mustbe further expanded and improved making it suitable forpublic-participation decision-making problems underchanging scenarios Second according to the settingprinciple of reference points the interval range of the at-tribute reference points affects the decision-making resultse interval range of the attribute reference points is largeand the discrimination of the evaluation results for dif-ferent alternatives is less obvious Because the intervalrange of attribute reference points is set by the compre-hensive ambiguity of public expectations if the dispersiondegree of public expectations is overly high (ie thecomprehensive ambiguity of public expectations is overlyhigh) the interval range of attribute reference points is alsoset over wide corresponding values is outcome may leadto a lack of differentiation in the decision results ereforeto ensure the effectiveness of the decision-making resultsreasonable interval ranges must be set for attribute refer-ence points according to the characteristics of public ex-pectation information which is also worth an in-depthstudy

Data Availability

e data used to support the findings of this study are in-cluded within the manuscript

Table 10 Decision results for different conditions of reference points

θ Reference points Comprehensive prospect value Alternatives ranking

0 1113957rlowast1 [44600 44600] 1113957r

lowast2 [6949 6949]

1113957rlowast3 [2899000 2899000] 1113957r

lowast4 [075 075]

V1 04365

V2 06769

V3 04699V2 gtV3 gtV1

025 1113957rlowast1 [44292 44909] 1113957r

lowast2 [6900 7000]

1113957rlowast3 [2873845 2924155] 1113957r

lowast4 [074 077]

V1 04138

V2 06068

V3 04442V2 gtV3 gtV1

050 1113957rlowast1 [43983 45217] 1113957r

lowast2 [6851 7050]

1113957rlowast3 [2848690 2949311] 1113957r

lowast4 [072 078]

V1 04104

V2 05425

V3 04231V2 gtV3 gtV1

075 1113957rlowast1 [43675 45526] 1113957r

lowast2 [6801 7099]

1113957rlowast3 [2823534 2974466] 1113957r

lowast4 [071 080]

V1 04141

V2 04827

V3 04066V2 gtV1 gtV3

1 1113957rlowast1 [43367 45834] 1113957r

lowast2 [6750 7148]

1113957rlowast3 [2798379 2999621] 1113957r

lowast4 [069 081]

V1 03994

V2 04211

V3 03816V2 gtV1 gtV3

12 Mathematical Problems in Engineering

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is research was funded by the National Natural ScienceFoundation of China (no 71902058) Natural ScienceFoundation of Hunan Province (nos 2018JJ3617 and2021JJ41088) Social Science Foundation of Hunan Province(no 19YBQ113) Scientific Research Foundation of HunanEducation Department (nos 18B484 19A06 and 20B155)and Social Science Achievements Appraisal CommitteeFoundation of Hunan Provincial Department (noXSP21YBC218)

References

[1] S Song Z Guo and X Wang ldquoRetracted article the cor-relation between social transformation economic risk andinternet public opinionrdquo Behaviour amp Information Technol-ogy vol 40 no 7 pp 723ndash733 2020

[2] W Chen F Tu and P Zheng ldquoA transnational networkedpublic sphere of air pollution analysis of a Twitter network ofPM25 from the risk society perspectiverdquo InformationCommunication amp Society vol 20 no 7 pp 1005ndash1023 2017

[3] X Yao J He and C Bao ldquoPublic participation modes inChinarsquos environmental impact assessment process an ana-lytical framework based on participation extent and conflictlevelrdquo Environmental Impact Assessment Review vol 84p 106400 2020

[4] X Zhang J G Xu and Y Ju ldquoPublic participation in NIMBYrisk mitigation A discourse zoning approach in the Chinesecontextrdquo Land Use Policy vol 77 pp 559ndash575 2018

[5] T Webler and S Tuler ldquoFour decades of public participationin risk decision makingrdquo Risk Analysis vol 41 no 3pp 503ndash518 2021

[6] J P Voszlig and N Amelung ldquoInnovating public participationmethods Technoscientization and reflexive engagementrdquoSocial Studies of Science vol 46 no 5 pp 749ndash772 2016

[7] A M Rıos B Benito and F Bastida ldquoFactors explainingpublic participation in the central government budget pro-cessrdquo Australian Journal of Public Administration vol 76no 1 pp 48ndash64 2017

[8] Y Zhou L Hou Y Yang H-Y Chong and S Moon ldquoAcomparative review and framework development on publicparticipation for decision-making in Chinese public projectsrdquoEnvironmental Impact Assessment Review vol 75 pp 79ndash872019

[9] T H You J Zhang and Z P Fan ldquoMulti-attribute onlinereview decision making method based on sentiment analysisand evidence theoryrdquo Journal of Systems amp Managementvol 28 no 3 pp 536ndash544 2019 in Chinese

[10] R L Charney T Rebmann P Dalawari and A EndrizalldquoPublic expectations of hospitals to provide resources andservices to the uninjured during disasters A qualitativestudyrdquo Health Security vol 14 no 6 pp 389ndash396 2016

[11] A Yildiz E Ayyildiz A Taskin Gumus and C Ozkan ldquoAframework to prioritize the public expectations from watertreatment plants based on trapezoidal type-2 fuzzy ahpmethodrdquo Environmental Management vol 67 no 3pp 439ndash448 2020

[12] Z Zhang Y Gao and Z L Li ldquoConsensus reaching for socialnetwork group decision making by considering leadershipand bounded confidencerdquo Knowledge-Based Systems vol 204pp 1ndash12 2020

[13] J J Zhu Z Z Ma H H Wang and Y Chen ldquoRisk decision-making method using interval numbers and its applicationbased on the prospect value with multiple reference pointsrdquoInformation Sciences vol 12 no 3 pp 385-386 2017

[14] X Li and X Chen ldquoValue determination method based onmultiple reference points under a trapezoidal intuitionisticfuzzy environmentrdquo Applied Soft Computing vol 63pp 39ndash49 2018

[15] J Gao Z Xu and H Liao ldquoA dynamic reference pointmethod for emergency response under hesitant probabilisticfuzzy environmentrdquo International Journal of Fuzzy Systemsvol 19 no 5 pp 1261ndash1278 2017

[16] C H Li W Li M J Li et al ldquoTarget-oriented model andapproach for attribute value evaluation with multiple refer-ence pointsrdquo Chinese Journal of Management Science vol 25no 7 pp 163ndash175 2017 in Chinese

[17] E Mastrocinque F J Ramırez A Honrubia-Escribano andD T Pham ldquoAn AHP-based multi-criteria model for sus-tainable supply chain development in the renewable energysectorrdquo Expert Systems with Applications vol 150 pp 1ndash172020

[18] L Chen Z Li and X Deng ldquoEmergency alternative evalu-ation under group decision makers a new method based onentropy weight and dematelrdquo International Journal of SystemsScience vol 51 no 3 pp 570ndash583 2020

[19] X KWang Y TWang J QWang P F Cheng and L Li ldquoATODIM-PROMETHEE II based multi-criteria group decisionmaking method for risk evaluation of water resource carryingcapacity under probabilistic linguistic Z-number circum-stancesrdquo Mathematics vol 8 no 7 p 1190 2020

[20] Y Liu Y Dong H Liang F Chiclana and E Herrera-Viedma ldquoMultiple attribute strategic weight manipulationwith minimum cost in a group decision making context withinterval attribute weights informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 49 no 10pp 1981ndash1992 2018

[21] F Meng C Tan and X Chen ldquoAn approach to Atanassovrsquosinterval-valued intuitionistic fuzzy multi-attribute decisionmaking based on prospect theoryrdquo International Journal ofComputational Intelligence Systems vol 8 no 3 pp 591ndash6052015

[22] N Zarbakhshnia Y Wu K Govindan and H Soleimani ldquoAnovel hybrid multiple attribute decision-making approach foroutsourcing sustainable reverse logisticsrdquo Journal of CleanerProduction vol 242 Article ID 118461 2020

[23] J J H Liou Y C Chuang E K Zavadskas and G H TzengldquoData-driven hybrid multiple attribute decision-makingmodel for green supplier evaluation and performance im-provementrdquo Journal of Cleaner Production vol 241 ArticleID 118321 2019

[24] Y H Pan and X L Geng ldquoHybrid multiple attribute decisionmaking approach based on Mo-RVIKORrdquo Chinese Journal ofManagement Science vol 27 no 12 pp 143ndash151 2019

[25] M Akram and D Shumaiza ldquoMulti-criteria decision makingbased on q-rung orthopair fuzzy promethee approachrdquo Ira-nian Journal of Fuzzy Systems vol 18 no 5 pp 107ndash127 2021

[26] C Jana G Muhiuddin and M Pal ldquoMulti-criteria decisionmaking approach based on SVTrN Dombi aggregationfunctionsrdquo Artificial Intelligence Review vol 54 no 5pp 3685ndash3723 2021

Mathematical Problems in Engineering 13

[27] C Jana G Muhiuddin and M Pal ldquoMultiple-attribute de-cision making problems based on SVTNH methodsrdquo Journalof Ambient Intelligence and Humanized Computing vol 11no 9 pp 3717ndash3733 2020

[28] C Jana GMuhiuddin andM Pal ldquoSomeDombi aggregationof Q -rung orthopair fuzzy numbers in multiple-attributedecision makingrdquo International Journal of Intelligent Systemsvol 34 no 12 pp 3220ndash3240 2019

[29] G Shahzadi G Muhiuddin M Arif Butt and A AshrafldquoHamacher interactive hybrid weighted averaging operatorsunder fermatean fuzzy numbersrdquo Journal of Mathematicsvol 2021 no 10 17 pages Article ID 5556017 2021

[30] K Zhang J Zhan and X Wang ldquoTOPSIS-WAA methodbased on a covering-based fuzzy rough set An application torating problemrdquo Information Sciences vol 539 pp 397ndash4212020

[31] S F Huang ldquoUsing Linguistic VIKOR and fuzzy cognitivemaps to select virtual reality games development projectrdquoMathematics vol 9 no 11 p 1253 2021

[32] M Akram A Luqman and C Kahraman ldquoHesitant py-thagorean fuzzy ELECTRE-II method for multi-criteria de-cision-making problemsrdquo Applied Soft Computing vol 108Article ID 107479 2021

[33] C Erdin and G Ozkaya ldquoTurkeyrsquos 2023 energy strategies andinvestment opportunities for renewable energy sources Siteselection based on ELECTRErdquo Sustainability vol 11 no 7Article ID 2136 2019

[34] T Wang H Li X Zhou D Liu and B Huang ldquoree-waydecision based on third-generation prospect theory with Z-numbersrdquo Information Sciences vol 569 pp 13ndash38 2021

[35] Z J Du S M Yu H Y Luo and X D Lin ldquoConsensusconvergence in large-group social network environmentcoordination between trust relationship and opinion simi-larityrdquo Knowledge-Based Systems vol 217 Article ID 1068282021

[36] J Wu F Chiclana H Fujita and E Herrera-Viedma ldquoAvisual interaction consensus model for social network groupdecision making with trust propagationrdquo Knowledge-BasedSystems vol 122 pp 39ndash50 2017

[37] L X Chen and N F Luo ldquoPythagorean fuzzy multi-criteriadecision-making based on prospect theoryrdquo Systems Engi-neering Keory amp Practice vol 40 no 3 pp 726ndash735 2020 inChinese

[38] Z S Xu Linguistic Decision Making Springer-Verlag BerlinGermany 2012

[39] R E Moore Methods and Applications of Interval AnalysisPrentice-Hall Hoboken NY USA 1979

[40] R R Ren W W Li M Zhao and X Li ldquoA large groupdecision making method based on public evaluationrdquoManagement Review vol 30 no 10 pp 238ndash247 2018

[41] L Wang Y M Wang and L Martınez ldquoA group decisionmethod based on prospect theory for emergency situationsrdquoInformation Sciences vol 418-419 pp 119ndash135 2017

[42] Y Wu C Xu and T Zhang ldquoEvaluation of renewable powersources using a fuzzy MCDM based on cumulative prospecttheory a case in Chinardquo Energy vol 147 pp 1227ndash1239 2018

[43] M M Salih B B Zaidan A A Zaidan and M A AhmedldquoSurvey on fuzzy TOPSIS state-of-the-art between 2007 and2017rdquo Computers amp Operations Research vol 104 pp 207ndash227 2019

[44] H Xiao Y Zhang X Liu H Yin P Liu and D C Liu ldquoArapid ultrasound vascular disease screening method usingPauTa Criterionrdquo Journal of Physics Conference Seriesvol 1769 Article ID 012009 2021

14 Mathematical Problems in Engineering

inconsistent with the results obtained in this studyemainreason for the inconsistency of decision results is that thereference point setting in each decision method is different

e setting principle of reference points for the twoexisting methods is that reference points are obtainedaccording to the public expectation or the attribute value of thealternative e setting principle for the reference points usedin the two existing methods is relatively simple but it does notconsider the characteristics of public expectations For deci-sion-making activities in major public affairs public expec-tations must be fully considered to ensure the effectiveness ofthe decision-making results erefore public expectationsmust be considered when setting reference points

Based on the above analysis we develop the settingprinciple for reference points by considering public ex-pectations First the information on public expectations isdiscrete and has the characteristics of a normal distributionthus the mean value of public expectations is taken as thestandard setting for the reference point so that the referencepoint can effectively reflect the expectations of the publicgroup Second due to the uncertainty of the expectationevaluation by public individuals the expected value isusually expressed in an interval number which reflects theambiguity of public expectations e comprehensive ex-pected ambiguity of the public is used to represent theuncertain characteristics of public expectations e refer-ence point is set by the mean value of public expectationsand the comprehensive expected ambiguity of the public tobetter reflect the actual situation of public expectation ex-pression e results of the reference points are presented asinterval numbers e value of the reference points reflectsthe expectations of the public group and considers theuncertainty of the public evaluation

If an attribute value in interval form falls within therange of the corresponding reference points in whole or inpart we consider that part of the attribute value that fallswithin the range of the reference points exactly meetspublic expectations which means that the prospect valuefor this part of the attribute is zero erefore when theprospect value is calculated based on the interval reference

point the inclusion or cross relationship between theattribute interval and corresponding reference point in-terval should be considered For example Table 8 indicatesthat the normalized values of attribute 3 for alternatives 1and 3 are p13 [00318 02734] and p33 [00318

04179] respectively Moreover p13 and p33 are not equaland the normalized reference point for attribute 3 isq3 [00000 04353] It is apparent that p13 and p33 areentirely within the interval range of q3 We can affirm thatthe prospect values of attribute 3 for alternatives 1 and 3are both zero (v13 v33 0) In addition we use equations(7) and (8) and Table 8 to calculate the prospect values ofp13 and p33 we can also obtain the same result If wechoose the double-reference point decision-makingmethod to calculate the prospect value of p13 and p33 theresults of the prospect value for p13 and p33 arev13 minus 07329 and v33 minus 06732 which is different fromthe result of the proposed method erefore a differentsetting principle for the reference point leads to differentdecision-making results explaining why the sorting resultsare different in Table 9 erefore if we need to set ref-erence points for decision-making we must set themaccording to the actual situation for the decision problemand decision requirements to guarantee the effectiveness ofthe decision results

52 Sensitivity Analysis of the Reference Point Interval RangeAccording to the content above the comprehensive ambi-guity of public expectations is related to the interval range ofattribute reference points affecting the decision-makingresultse interval adjustment coefficient of reference pointθ (0le θle 1) is introduced in the expression of referencepoints to study the relationship between the value range ofthe reference points and decision results further e ref-erence point of attribute j is defined as1113957rlowastj [1113957rlowastLj 1113957rlowastUj ] [μ(rk

j ) minus θcj μ(rkj) + θcj] e compre-

hensive ambiguity of public expectations for each attribute cj

is known When the values of θ are 0 025 05 075 and 1the corresponding value range of reference points changes

Table 9 Alternative ranking results of different decision methods

Methods Attributereference points Decision results Ranking of

the alternatives

Double-reference point decision-makingmethod based on prospect theory

1113957rlowast+1 [22277 22277] 1113957r

lowast+2 [4405 4405]

1113957rlowast+3 [2029300 2029300] 1113957r

lowast+4 [100 100] V1 minus 01738

V2 minus 00983

V3 minus 01654

V2 gtV3 gtV11113957rlowastminus1 [66923 66923] 1113957r

lowastminus2 [9493 9493]

1113957rlowastminus3 [3768700 3768700] 1113957r

lowastminus4 [000 000]

TOPSIS

1113957rlowast+1 [33000 33000] 1113957r

lowast+2 [6100 6100]

1113957rlowast+3 [2537300 2537300] 1113957r

lowast+4 [0875 0875] V1 05719

V2 04772

V3 05564

V1 gtV3 gtV21113957rlowastminus1 [37900 37900] 1113957r

lowastminus2 [7100 7100]

1113957rlowastminus3 [2984900 2984900] 1113957r

lowastminus4 [075 075]

e proposed method 1113957rlowast1 [43367 45834] 1113957r

lowast2 [6750 7148]

1113957rlowast3 [2798379 2999621] 1113957r

lowast4 [069 081]

V1 03994

V2 04211

V3 03816

V2 gtV1 gtV3

Mathematical Problems in Engineering 11

accordingly e decision results for different conditions ofreference points are presented in Table 10

As listed in Table 10 when the interval adjustmentcoefficient of the reference points increases gradually theinterval range of the reference points for different attributesis also constantly enlarged so the alternative ranking resultsare not the same As observed in Table 10 if θ isin [0 05] thesorting result is V2 gtV3 gtV1 Moreover if θ isin [075 1] thesorting result is V2 gtV1 gtV3 In addition the alternativeranking results in Table 10 with the continuous expansion ofthe interval for reference points indicate that the compre-hensive prospect value of each alternative also constantlydecreases In addition the differentiation degree of thecomprehensive prospect value between alternatives alsogradually decreases erefore to ensure the effectiveness ofsetting attribute reference points we must consider theinfluence of the interval range of the reference points for thedecision results to reduce the difficulty of decision-makingand improve the accuracy of decision results

6 Conclusion

is paper proposed a new decision-making approach for amixed multiattribute decision-making problem with un-known attribute weights e advantages of this approachare summarized as follows First the reference point for eachattribute is set based on the distribution and comprehensiveambiguity of public expectations making the attributereference points better reflect the publicrsquos expected groupopinions and expectation uncertainty e effectiveness ofthe decision result is guaranteed Second in solving theattribute weights the attribute importance given by thepublic is used to determine the value range of the attributeweights so that the weighting results are in accordance withpublic opinions making the results of the attribute weightmore acceptable en the exact values of the attributeweights are determined to maximize the attribute infor-mation deviation of all the alternatives improving thediscrimination of alternatives ird each alternative is

evaluated based on the prospect theory to satisfy the cal-culation needs of the prospect valuee different expressionforms of the attribute information and reference point areunified and normalization is performed e operationabove can eliminate the influence of the expression form anddimension on decision-making making the decision op-erations smoother

e proposed approach also has certain limitationsFirst in the actual decision-making process the decision-making scenarios in some decision-making problems arenot static Changes in the decision-making scenarios easilycause decision risk which negatively affects the decisionactivity Our proposed approach does not consider dy-namic decision scenarios so the proposed approach mustbe further expanded and improved making it suitable forpublic-participation decision-making problems underchanging scenarios Second according to the settingprinciple of reference points the interval range of the at-tribute reference points affects the decision-making resultse interval range of the attribute reference points is largeand the discrimination of the evaluation results for dif-ferent alternatives is less obvious Because the intervalrange of attribute reference points is set by the compre-hensive ambiguity of public expectations if the dispersiondegree of public expectations is overly high (ie thecomprehensive ambiguity of public expectations is overlyhigh) the interval range of attribute reference points is alsoset over wide corresponding values is outcome may leadto a lack of differentiation in the decision results ereforeto ensure the effectiveness of the decision-making resultsreasonable interval ranges must be set for attribute refer-ence points according to the characteristics of public ex-pectation information which is also worth an in-depthstudy

Data Availability

e data used to support the findings of this study are in-cluded within the manuscript

Table 10 Decision results for different conditions of reference points

θ Reference points Comprehensive prospect value Alternatives ranking

0 1113957rlowast1 [44600 44600] 1113957r

lowast2 [6949 6949]

1113957rlowast3 [2899000 2899000] 1113957r

lowast4 [075 075]

V1 04365

V2 06769

V3 04699V2 gtV3 gtV1

025 1113957rlowast1 [44292 44909] 1113957r

lowast2 [6900 7000]

1113957rlowast3 [2873845 2924155] 1113957r

lowast4 [074 077]

V1 04138

V2 06068

V3 04442V2 gtV3 gtV1

050 1113957rlowast1 [43983 45217] 1113957r

lowast2 [6851 7050]

1113957rlowast3 [2848690 2949311] 1113957r

lowast4 [072 078]

V1 04104

V2 05425

V3 04231V2 gtV3 gtV1

075 1113957rlowast1 [43675 45526] 1113957r

lowast2 [6801 7099]

1113957rlowast3 [2823534 2974466] 1113957r

lowast4 [071 080]

V1 04141

V2 04827

V3 04066V2 gtV1 gtV3

1 1113957rlowast1 [43367 45834] 1113957r

lowast2 [6750 7148]

1113957rlowast3 [2798379 2999621] 1113957r

lowast4 [069 081]

V1 03994

V2 04211

V3 03816V2 gtV1 gtV3

12 Mathematical Problems in Engineering

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is research was funded by the National Natural ScienceFoundation of China (no 71902058) Natural ScienceFoundation of Hunan Province (nos 2018JJ3617 and2021JJ41088) Social Science Foundation of Hunan Province(no 19YBQ113) Scientific Research Foundation of HunanEducation Department (nos 18B484 19A06 and 20B155)and Social Science Achievements Appraisal CommitteeFoundation of Hunan Provincial Department (noXSP21YBC218)

References

[1] S Song Z Guo and X Wang ldquoRetracted article the cor-relation between social transformation economic risk andinternet public opinionrdquo Behaviour amp Information Technol-ogy vol 40 no 7 pp 723ndash733 2020

[2] W Chen F Tu and P Zheng ldquoA transnational networkedpublic sphere of air pollution analysis of a Twitter network ofPM25 from the risk society perspectiverdquo InformationCommunication amp Society vol 20 no 7 pp 1005ndash1023 2017

[3] X Yao J He and C Bao ldquoPublic participation modes inChinarsquos environmental impact assessment process an ana-lytical framework based on participation extent and conflictlevelrdquo Environmental Impact Assessment Review vol 84p 106400 2020

[4] X Zhang J G Xu and Y Ju ldquoPublic participation in NIMBYrisk mitigation A discourse zoning approach in the Chinesecontextrdquo Land Use Policy vol 77 pp 559ndash575 2018

[5] T Webler and S Tuler ldquoFour decades of public participationin risk decision makingrdquo Risk Analysis vol 41 no 3pp 503ndash518 2021

[6] J P Voszlig and N Amelung ldquoInnovating public participationmethods Technoscientization and reflexive engagementrdquoSocial Studies of Science vol 46 no 5 pp 749ndash772 2016

[7] A M Rıos B Benito and F Bastida ldquoFactors explainingpublic participation in the central government budget pro-cessrdquo Australian Journal of Public Administration vol 76no 1 pp 48ndash64 2017

[8] Y Zhou L Hou Y Yang H-Y Chong and S Moon ldquoAcomparative review and framework development on publicparticipation for decision-making in Chinese public projectsrdquoEnvironmental Impact Assessment Review vol 75 pp 79ndash872019

[9] T H You J Zhang and Z P Fan ldquoMulti-attribute onlinereview decision making method based on sentiment analysisand evidence theoryrdquo Journal of Systems amp Managementvol 28 no 3 pp 536ndash544 2019 in Chinese

[10] R L Charney T Rebmann P Dalawari and A EndrizalldquoPublic expectations of hospitals to provide resources andservices to the uninjured during disasters A qualitativestudyrdquo Health Security vol 14 no 6 pp 389ndash396 2016

[11] A Yildiz E Ayyildiz A Taskin Gumus and C Ozkan ldquoAframework to prioritize the public expectations from watertreatment plants based on trapezoidal type-2 fuzzy ahpmethodrdquo Environmental Management vol 67 no 3pp 439ndash448 2020

[12] Z Zhang Y Gao and Z L Li ldquoConsensus reaching for socialnetwork group decision making by considering leadershipand bounded confidencerdquo Knowledge-Based Systems vol 204pp 1ndash12 2020

[13] J J Zhu Z Z Ma H H Wang and Y Chen ldquoRisk decision-making method using interval numbers and its applicationbased on the prospect value with multiple reference pointsrdquoInformation Sciences vol 12 no 3 pp 385-386 2017

[14] X Li and X Chen ldquoValue determination method based onmultiple reference points under a trapezoidal intuitionisticfuzzy environmentrdquo Applied Soft Computing vol 63pp 39ndash49 2018

[15] J Gao Z Xu and H Liao ldquoA dynamic reference pointmethod for emergency response under hesitant probabilisticfuzzy environmentrdquo International Journal of Fuzzy Systemsvol 19 no 5 pp 1261ndash1278 2017

[16] C H Li W Li M J Li et al ldquoTarget-oriented model andapproach for attribute value evaluation with multiple refer-ence pointsrdquo Chinese Journal of Management Science vol 25no 7 pp 163ndash175 2017 in Chinese

[17] E Mastrocinque F J Ramırez A Honrubia-Escribano andD T Pham ldquoAn AHP-based multi-criteria model for sus-tainable supply chain development in the renewable energysectorrdquo Expert Systems with Applications vol 150 pp 1ndash172020

[18] L Chen Z Li and X Deng ldquoEmergency alternative evalu-ation under group decision makers a new method based onentropy weight and dematelrdquo International Journal of SystemsScience vol 51 no 3 pp 570ndash583 2020

[19] X KWang Y TWang J QWang P F Cheng and L Li ldquoATODIM-PROMETHEE II based multi-criteria group decisionmaking method for risk evaluation of water resource carryingcapacity under probabilistic linguistic Z-number circum-stancesrdquo Mathematics vol 8 no 7 p 1190 2020

[20] Y Liu Y Dong H Liang F Chiclana and E Herrera-Viedma ldquoMultiple attribute strategic weight manipulationwith minimum cost in a group decision making context withinterval attribute weights informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 49 no 10pp 1981ndash1992 2018

[21] F Meng C Tan and X Chen ldquoAn approach to Atanassovrsquosinterval-valued intuitionistic fuzzy multi-attribute decisionmaking based on prospect theoryrdquo International Journal ofComputational Intelligence Systems vol 8 no 3 pp 591ndash6052015

[22] N Zarbakhshnia Y Wu K Govindan and H Soleimani ldquoAnovel hybrid multiple attribute decision-making approach foroutsourcing sustainable reverse logisticsrdquo Journal of CleanerProduction vol 242 Article ID 118461 2020

[23] J J H Liou Y C Chuang E K Zavadskas and G H TzengldquoData-driven hybrid multiple attribute decision-makingmodel for green supplier evaluation and performance im-provementrdquo Journal of Cleaner Production vol 241 ArticleID 118321 2019

[24] Y H Pan and X L Geng ldquoHybrid multiple attribute decisionmaking approach based on Mo-RVIKORrdquo Chinese Journal ofManagement Science vol 27 no 12 pp 143ndash151 2019

[25] M Akram and D Shumaiza ldquoMulti-criteria decision makingbased on q-rung orthopair fuzzy promethee approachrdquo Ira-nian Journal of Fuzzy Systems vol 18 no 5 pp 107ndash127 2021

[26] C Jana G Muhiuddin and M Pal ldquoMulti-criteria decisionmaking approach based on SVTrN Dombi aggregationfunctionsrdquo Artificial Intelligence Review vol 54 no 5pp 3685ndash3723 2021

Mathematical Problems in Engineering 13

[27] C Jana G Muhiuddin and M Pal ldquoMultiple-attribute de-cision making problems based on SVTNH methodsrdquo Journalof Ambient Intelligence and Humanized Computing vol 11no 9 pp 3717ndash3733 2020

[28] C Jana GMuhiuddin andM Pal ldquoSomeDombi aggregationof Q -rung orthopair fuzzy numbers in multiple-attributedecision makingrdquo International Journal of Intelligent Systemsvol 34 no 12 pp 3220ndash3240 2019

[29] G Shahzadi G Muhiuddin M Arif Butt and A AshrafldquoHamacher interactive hybrid weighted averaging operatorsunder fermatean fuzzy numbersrdquo Journal of Mathematicsvol 2021 no 10 17 pages Article ID 5556017 2021

[30] K Zhang J Zhan and X Wang ldquoTOPSIS-WAA methodbased on a covering-based fuzzy rough set An application torating problemrdquo Information Sciences vol 539 pp 397ndash4212020

[31] S F Huang ldquoUsing Linguistic VIKOR and fuzzy cognitivemaps to select virtual reality games development projectrdquoMathematics vol 9 no 11 p 1253 2021

[32] M Akram A Luqman and C Kahraman ldquoHesitant py-thagorean fuzzy ELECTRE-II method for multi-criteria de-cision-making problemsrdquo Applied Soft Computing vol 108Article ID 107479 2021

[33] C Erdin and G Ozkaya ldquoTurkeyrsquos 2023 energy strategies andinvestment opportunities for renewable energy sources Siteselection based on ELECTRErdquo Sustainability vol 11 no 7Article ID 2136 2019

[34] T Wang H Li X Zhou D Liu and B Huang ldquoree-waydecision based on third-generation prospect theory with Z-numbersrdquo Information Sciences vol 569 pp 13ndash38 2021

[35] Z J Du S M Yu H Y Luo and X D Lin ldquoConsensusconvergence in large-group social network environmentcoordination between trust relationship and opinion simi-larityrdquo Knowledge-Based Systems vol 217 Article ID 1068282021

[36] J Wu F Chiclana H Fujita and E Herrera-Viedma ldquoAvisual interaction consensus model for social network groupdecision making with trust propagationrdquo Knowledge-BasedSystems vol 122 pp 39ndash50 2017

[37] L X Chen and N F Luo ldquoPythagorean fuzzy multi-criteriadecision-making based on prospect theoryrdquo Systems Engi-neering Keory amp Practice vol 40 no 3 pp 726ndash735 2020 inChinese

[38] Z S Xu Linguistic Decision Making Springer-Verlag BerlinGermany 2012

[39] R E Moore Methods and Applications of Interval AnalysisPrentice-Hall Hoboken NY USA 1979

[40] R R Ren W W Li M Zhao and X Li ldquoA large groupdecision making method based on public evaluationrdquoManagement Review vol 30 no 10 pp 238ndash247 2018

[41] L Wang Y M Wang and L Martınez ldquoA group decisionmethod based on prospect theory for emergency situationsrdquoInformation Sciences vol 418-419 pp 119ndash135 2017

[42] Y Wu C Xu and T Zhang ldquoEvaluation of renewable powersources using a fuzzy MCDM based on cumulative prospecttheory a case in Chinardquo Energy vol 147 pp 1227ndash1239 2018

[43] M M Salih B B Zaidan A A Zaidan and M A AhmedldquoSurvey on fuzzy TOPSIS state-of-the-art between 2007 and2017rdquo Computers amp Operations Research vol 104 pp 207ndash227 2019

[44] H Xiao Y Zhang X Liu H Yin P Liu and D C Liu ldquoArapid ultrasound vascular disease screening method usingPauTa Criterionrdquo Journal of Physics Conference Seriesvol 1769 Article ID 012009 2021

14 Mathematical Problems in Engineering

accordingly e decision results for different conditions ofreference points are presented in Table 10

As listed in Table 10 when the interval adjustmentcoefficient of the reference points increases gradually theinterval range of the reference points for different attributesis also constantly enlarged so the alternative ranking resultsare not the same As observed in Table 10 if θ isin [0 05] thesorting result is V2 gtV3 gtV1 Moreover if θ isin [075 1] thesorting result is V2 gtV1 gtV3 In addition the alternativeranking results in Table 10 with the continuous expansion ofthe interval for reference points indicate that the compre-hensive prospect value of each alternative also constantlydecreases In addition the differentiation degree of thecomprehensive prospect value between alternatives alsogradually decreases erefore to ensure the effectiveness ofsetting attribute reference points we must consider theinfluence of the interval range of the reference points for thedecision results to reduce the difficulty of decision-makingand improve the accuracy of decision results

6 Conclusion

is paper proposed a new decision-making approach for amixed multiattribute decision-making problem with un-known attribute weights e advantages of this approachare summarized as follows First the reference point for eachattribute is set based on the distribution and comprehensiveambiguity of public expectations making the attributereference points better reflect the publicrsquos expected groupopinions and expectation uncertainty e effectiveness ofthe decision result is guaranteed Second in solving theattribute weights the attribute importance given by thepublic is used to determine the value range of the attributeweights so that the weighting results are in accordance withpublic opinions making the results of the attribute weightmore acceptable en the exact values of the attributeweights are determined to maximize the attribute infor-mation deviation of all the alternatives improving thediscrimination of alternatives ird each alternative is

evaluated based on the prospect theory to satisfy the cal-culation needs of the prospect valuee different expressionforms of the attribute information and reference point areunified and normalization is performed e operationabove can eliminate the influence of the expression form anddimension on decision-making making the decision op-erations smoother

e proposed approach also has certain limitationsFirst in the actual decision-making process the decision-making scenarios in some decision-making problems arenot static Changes in the decision-making scenarios easilycause decision risk which negatively affects the decisionactivity Our proposed approach does not consider dy-namic decision scenarios so the proposed approach mustbe further expanded and improved making it suitable forpublic-participation decision-making problems underchanging scenarios Second according to the settingprinciple of reference points the interval range of the at-tribute reference points affects the decision-making resultse interval range of the attribute reference points is largeand the discrimination of the evaluation results for dif-ferent alternatives is less obvious Because the intervalrange of attribute reference points is set by the compre-hensive ambiguity of public expectations if the dispersiondegree of public expectations is overly high (ie thecomprehensive ambiguity of public expectations is overlyhigh) the interval range of attribute reference points is alsoset over wide corresponding values is outcome may leadto a lack of differentiation in the decision results ereforeto ensure the effectiveness of the decision-making resultsreasonable interval ranges must be set for attribute refer-ence points according to the characteristics of public ex-pectation information which is also worth an in-depthstudy

Data Availability

e data used to support the findings of this study are in-cluded within the manuscript

Table 10 Decision results for different conditions of reference points

θ Reference points Comprehensive prospect value Alternatives ranking

0 1113957rlowast1 [44600 44600] 1113957r

lowast2 [6949 6949]

1113957rlowast3 [2899000 2899000] 1113957r

lowast4 [075 075]

V1 04365

V2 06769

V3 04699V2 gtV3 gtV1

025 1113957rlowast1 [44292 44909] 1113957r

lowast2 [6900 7000]

1113957rlowast3 [2873845 2924155] 1113957r

lowast4 [074 077]

V1 04138

V2 06068

V3 04442V2 gtV3 gtV1

050 1113957rlowast1 [43983 45217] 1113957r

lowast2 [6851 7050]

1113957rlowast3 [2848690 2949311] 1113957r

lowast4 [072 078]

V1 04104

V2 05425

V3 04231V2 gtV3 gtV1

075 1113957rlowast1 [43675 45526] 1113957r

lowast2 [6801 7099]

1113957rlowast3 [2823534 2974466] 1113957r

lowast4 [071 080]

V1 04141

V2 04827

V3 04066V2 gtV1 gtV3

1 1113957rlowast1 [43367 45834] 1113957r

lowast2 [6750 7148]

1113957rlowast3 [2798379 2999621] 1113957r

lowast4 [069 081]

V1 03994

V2 04211

V3 03816V2 gtV1 gtV3

12 Mathematical Problems in Engineering

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is research was funded by the National Natural ScienceFoundation of China (no 71902058) Natural ScienceFoundation of Hunan Province (nos 2018JJ3617 and2021JJ41088) Social Science Foundation of Hunan Province(no 19YBQ113) Scientific Research Foundation of HunanEducation Department (nos 18B484 19A06 and 20B155)and Social Science Achievements Appraisal CommitteeFoundation of Hunan Provincial Department (noXSP21YBC218)

References

[1] S Song Z Guo and X Wang ldquoRetracted article the cor-relation between social transformation economic risk andinternet public opinionrdquo Behaviour amp Information Technol-ogy vol 40 no 7 pp 723ndash733 2020

[2] W Chen F Tu and P Zheng ldquoA transnational networkedpublic sphere of air pollution analysis of a Twitter network ofPM25 from the risk society perspectiverdquo InformationCommunication amp Society vol 20 no 7 pp 1005ndash1023 2017

[3] X Yao J He and C Bao ldquoPublic participation modes inChinarsquos environmental impact assessment process an ana-lytical framework based on participation extent and conflictlevelrdquo Environmental Impact Assessment Review vol 84p 106400 2020

[4] X Zhang J G Xu and Y Ju ldquoPublic participation in NIMBYrisk mitigation A discourse zoning approach in the Chinesecontextrdquo Land Use Policy vol 77 pp 559ndash575 2018

[5] T Webler and S Tuler ldquoFour decades of public participationin risk decision makingrdquo Risk Analysis vol 41 no 3pp 503ndash518 2021

[6] J P Voszlig and N Amelung ldquoInnovating public participationmethods Technoscientization and reflexive engagementrdquoSocial Studies of Science vol 46 no 5 pp 749ndash772 2016

[7] A M Rıos B Benito and F Bastida ldquoFactors explainingpublic participation in the central government budget pro-cessrdquo Australian Journal of Public Administration vol 76no 1 pp 48ndash64 2017

[8] Y Zhou L Hou Y Yang H-Y Chong and S Moon ldquoAcomparative review and framework development on publicparticipation for decision-making in Chinese public projectsrdquoEnvironmental Impact Assessment Review vol 75 pp 79ndash872019

[9] T H You J Zhang and Z P Fan ldquoMulti-attribute onlinereview decision making method based on sentiment analysisand evidence theoryrdquo Journal of Systems amp Managementvol 28 no 3 pp 536ndash544 2019 in Chinese

[10] R L Charney T Rebmann P Dalawari and A EndrizalldquoPublic expectations of hospitals to provide resources andservices to the uninjured during disasters A qualitativestudyrdquo Health Security vol 14 no 6 pp 389ndash396 2016

[11] A Yildiz E Ayyildiz A Taskin Gumus and C Ozkan ldquoAframework to prioritize the public expectations from watertreatment plants based on trapezoidal type-2 fuzzy ahpmethodrdquo Environmental Management vol 67 no 3pp 439ndash448 2020

[12] Z Zhang Y Gao and Z L Li ldquoConsensus reaching for socialnetwork group decision making by considering leadershipand bounded confidencerdquo Knowledge-Based Systems vol 204pp 1ndash12 2020

[13] J J Zhu Z Z Ma H H Wang and Y Chen ldquoRisk decision-making method using interval numbers and its applicationbased on the prospect value with multiple reference pointsrdquoInformation Sciences vol 12 no 3 pp 385-386 2017

[14] X Li and X Chen ldquoValue determination method based onmultiple reference points under a trapezoidal intuitionisticfuzzy environmentrdquo Applied Soft Computing vol 63pp 39ndash49 2018

[15] J Gao Z Xu and H Liao ldquoA dynamic reference pointmethod for emergency response under hesitant probabilisticfuzzy environmentrdquo International Journal of Fuzzy Systemsvol 19 no 5 pp 1261ndash1278 2017

[16] C H Li W Li M J Li et al ldquoTarget-oriented model andapproach for attribute value evaluation with multiple refer-ence pointsrdquo Chinese Journal of Management Science vol 25no 7 pp 163ndash175 2017 in Chinese

[17] E Mastrocinque F J Ramırez A Honrubia-Escribano andD T Pham ldquoAn AHP-based multi-criteria model for sus-tainable supply chain development in the renewable energysectorrdquo Expert Systems with Applications vol 150 pp 1ndash172020

[18] L Chen Z Li and X Deng ldquoEmergency alternative evalu-ation under group decision makers a new method based onentropy weight and dematelrdquo International Journal of SystemsScience vol 51 no 3 pp 570ndash583 2020

[19] X KWang Y TWang J QWang P F Cheng and L Li ldquoATODIM-PROMETHEE II based multi-criteria group decisionmaking method for risk evaluation of water resource carryingcapacity under probabilistic linguistic Z-number circum-stancesrdquo Mathematics vol 8 no 7 p 1190 2020

[20] Y Liu Y Dong H Liang F Chiclana and E Herrera-Viedma ldquoMultiple attribute strategic weight manipulationwith minimum cost in a group decision making context withinterval attribute weights informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 49 no 10pp 1981ndash1992 2018

[21] F Meng C Tan and X Chen ldquoAn approach to Atanassovrsquosinterval-valued intuitionistic fuzzy multi-attribute decisionmaking based on prospect theoryrdquo International Journal ofComputational Intelligence Systems vol 8 no 3 pp 591ndash6052015

[22] N Zarbakhshnia Y Wu K Govindan and H Soleimani ldquoAnovel hybrid multiple attribute decision-making approach foroutsourcing sustainable reverse logisticsrdquo Journal of CleanerProduction vol 242 Article ID 118461 2020

[23] J J H Liou Y C Chuang E K Zavadskas and G H TzengldquoData-driven hybrid multiple attribute decision-makingmodel for green supplier evaluation and performance im-provementrdquo Journal of Cleaner Production vol 241 ArticleID 118321 2019

[24] Y H Pan and X L Geng ldquoHybrid multiple attribute decisionmaking approach based on Mo-RVIKORrdquo Chinese Journal ofManagement Science vol 27 no 12 pp 143ndash151 2019

[25] M Akram and D Shumaiza ldquoMulti-criteria decision makingbased on q-rung orthopair fuzzy promethee approachrdquo Ira-nian Journal of Fuzzy Systems vol 18 no 5 pp 107ndash127 2021

[26] C Jana G Muhiuddin and M Pal ldquoMulti-criteria decisionmaking approach based on SVTrN Dombi aggregationfunctionsrdquo Artificial Intelligence Review vol 54 no 5pp 3685ndash3723 2021

Mathematical Problems in Engineering 13

[27] C Jana G Muhiuddin and M Pal ldquoMultiple-attribute de-cision making problems based on SVTNH methodsrdquo Journalof Ambient Intelligence and Humanized Computing vol 11no 9 pp 3717ndash3733 2020

[28] C Jana GMuhiuddin andM Pal ldquoSomeDombi aggregationof Q -rung orthopair fuzzy numbers in multiple-attributedecision makingrdquo International Journal of Intelligent Systemsvol 34 no 12 pp 3220ndash3240 2019

[29] G Shahzadi G Muhiuddin M Arif Butt and A AshrafldquoHamacher interactive hybrid weighted averaging operatorsunder fermatean fuzzy numbersrdquo Journal of Mathematicsvol 2021 no 10 17 pages Article ID 5556017 2021

[30] K Zhang J Zhan and X Wang ldquoTOPSIS-WAA methodbased on a covering-based fuzzy rough set An application torating problemrdquo Information Sciences vol 539 pp 397ndash4212020

[31] S F Huang ldquoUsing Linguistic VIKOR and fuzzy cognitivemaps to select virtual reality games development projectrdquoMathematics vol 9 no 11 p 1253 2021

[32] M Akram A Luqman and C Kahraman ldquoHesitant py-thagorean fuzzy ELECTRE-II method for multi-criteria de-cision-making problemsrdquo Applied Soft Computing vol 108Article ID 107479 2021

[33] C Erdin and G Ozkaya ldquoTurkeyrsquos 2023 energy strategies andinvestment opportunities for renewable energy sources Siteselection based on ELECTRErdquo Sustainability vol 11 no 7Article ID 2136 2019

[34] T Wang H Li X Zhou D Liu and B Huang ldquoree-waydecision based on third-generation prospect theory with Z-numbersrdquo Information Sciences vol 569 pp 13ndash38 2021

[35] Z J Du S M Yu H Y Luo and X D Lin ldquoConsensusconvergence in large-group social network environmentcoordination between trust relationship and opinion simi-larityrdquo Knowledge-Based Systems vol 217 Article ID 1068282021

[36] J Wu F Chiclana H Fujita and E Herrera-Viedma ldquoAvisual interaction consensus model for social network groupdecision making with trust propagationrdquo Knowledge-BasedSystems vol 122 pp 39ndash50 2017

[37] L X Chen and N F Luo ldquoPythagorean fuzzy multi-criteriadecision-making based on prospect theoryrdquo Systems Engi-neering Keory amp Practice vol 40 no 3 pp 726ndash735 2020 inChinese

[38] Z S Xu Linguistic Decision Making Springer-Verlag BerlinGermany 2012

[39] R E Moore Methods and Applications of Interval AnalysisPrentice-Hall Hoboken NY USA 1979

[40] R R Ren W W Li M Zhao and X Li ldquoA large groupdecision making method based on public evaluationrdquoManagement Review vol 30 no 10 pp 238ndash247 2018

[41] L Wang Y M Wang and L Martınez ldquoA group decisionmethod based on prospect theory for emergency situationsrdquoInformation Sciences vol 418-419 pp 119ndash135 2017

[42] Y Wu C Xu and T Zhang ldquoEvaluation of renewable powersources using a fuzzy MCDM based on cumulative prospecttheory a case in Chinardquo Energy vol 147 pp 1227ndash1239 2018

[43] M M Salih B B Zaidan A A Zaidan and M A AhmedldquoSurvey on fuzzy TOPSIS state-of-the-art between 2007 and2017rdquo Computers amp Operations Research vol 104 pp 207ndash227 2019

[44] H Xiao Y Zhang X Liu H Yin P Liu and D C Liu ldquoArapid ultrasound vascular disease screening method usingPauTa Criterionrdquo Journal of Physics Conference Seriesvol 1769 Article ID 012009 2021

14 Mathematical Problems in Engineering

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is research was funded by the National Natural ScienceFoundation of China (no 71902058) Natural ScienceFoundation of Hunan Province (nos 2018JJ3617 and2021JJ41088) Social Science Foundation of Hunan Province(no 19YBQ113) Scientific Research Foundation of HunanEducation Department (nos 18B484 19A06 and 20B155)and Social Science Achievements Appraisal CommitteeFoundation of Hunan Provincial Department (noXSP21YBC218)

References

[1] S Song Z Guo and X Wang ldquoRetracted article the cor-relation between social transformation economic risk andinternet public opinionrdquo Behaviour amp Information Technol-ogy vol 40 no 7 pp 723ndash733 2020

[2] W Chen F Tu and P Zheng ldquoA transnational networkedpublic sphere of air pollution analysis of a Twitter network ofPM25 from the risk society perspectiverdquo InformationCommunication amp Society vol 20 no 7 pp 1005ndash1023 2017

[3] X Yao J He and C Bao ldquoPublic participation modes inChinarsquos environmental impact assessment process an ana-lytical framework based on participation extent and conflictlevelrdquo Environmental Impact Assessment Review vol 84p 106400 2020

[4] X Zhang J G Xu and Y Ju ldquoPublic participation in NIMBYrisk mitigation A discourse zoning approach in the Chinesecontextrdquo Land Use Policy vol 77 pp 559ndash575 2018

[5] T Webler and S Tuler ldquoFour decades of public participationin risk decision makingrdquo Risk Analysis vol 41 no 3pp 503ndash518 2021

[6] J P Voszlig and N Amelung ldquoInnovating public participationmethods Technoscientization and reflexive engagementrdquoSocial Studies of Science vol 46 no 5 pp 749ndash772 2016

[7] A M Rıos B Benito and F Bastida ldquoFactors explainingpublic participation in the central government budget pro-cessrdquo Australian Journal of Public Administration vol 76no 1 pp 48ndash64 2017

[8] Y Zhou L Hou Y Yang H-Y Chong and S Moon ldquoAcomparative review and framework development on publicparticipation for decision-making in Chinese public projectsrdquoEnvironmental Impact Assessment Review vol 75 pp 79ndash872019

[9] T H You J Zhang and Z P Fan ldquoMulti-attribute onlinereview decision making method based on sentiment analysisand evidence theoryrdquo Journal of Systems amp Managementvol 28 no 3 pp 536ndash544 2019 in Chinese

[10] R L Charney T Rebmann P Dalawari and A EndrizalldquoPublic expectations of hospitals to provide resources andservices to the uninjured during disasters A qualitativestudyrdquo Health Security vol 14 no 6 pp 389ndash396 2016

[11] A Yildiz E Ayyildiz A Taskin Gumus and C Ozkan ldquoAframework to prioritize the public expectations from watertreatment plants based on trapezoidal type-2 fuzzy ahpmethodrdquo Environmental Management vol 67 no 3pp 439ndash448 2020

[12] Z Zhang Y Gao and Z L Li ldquoConsensus reaching for socialnetwork group decision making by considering leadershipand bounded confidencerdquo Knowledge-Based Systems vol 204pp 1ndash12 2020

[13] J J Zhu Z Z Ma H H Wang and Y Chen ldquoRisk decision-making method using interval numbers and its applicationbased on the prospect value with multiple reference pointsrdquoInformation Sciences vol 12 no 3 pp 385-386 2017

[14] X Li and X Chen ldquoValue determination method based onmultiple reference points under a trapezoidal intuitionisticfuzzy environmentrdquo Applied Soft Computing vol 63pp 39ndash49 2018

[15] J Gao Z Xu and H Liao ldquoA dynamic reference pointmethod for emergency response under hesitant probabilisticfuzzy environmentrdquo International Journal of Fuzzy Systemsvol 19 no 5 pp 1261ndash1278 2017

[16] C H Li W Li M J Li et al ldquoTarget-oriented model andapproach for attribute value evaluation with multiple refer-ence pointsrdquo Chinese Journal of Management Science vol 25no 7 pp 163ndash175 2017 in Chinese

[17] E Mastrocinque F J Ramırez A Honrubia-Escribano andD T Pham ldquoAn AHP-based multi-criteria model for sus-tainable supply chain development in the renewable energysectorrdquo Expert Systems with Applications vol 150 pp 1ndash172020

[18] L Chen Z Li and X Deng ldquoEmergency alternative evalu-ation under group decision makers a new method based onentropy weight and dematelrdquo International Journal of SystemsScience vol 51 no 3 pp 570ndash583 2020

[19] X KWang Y TWang J QWang P F Cheng and L Li ldquoATODIM-PROMETHEE II based multi-criteria group decisionmaking method for risk evaluation of water resource carryingcapacity under probabilistic linguistic Z-number circum-stancesrdquo Mathematics vol 8 no 7 p 1190 2020

[20] Y Liu Y Dong H Liang F Chiclana and E Herrera-Viedma ldquoMultiple attribute strategic weight manipulationwith minimum cost in a group decision making context withinterval attribute weights informationrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 49 no 10pp 1981ndash1992 2018

[21] F Meng C Tan and X Chen ldquoAn approach to Atanassovrsquosinterval-valued intuitionistic fuzzy multi-attribute decisionmaking based on prospect theoryrdquo International Journal ofComputational Intelligence Systems vol 8 no 3 pp 591ndash6052015

[22] N Zarbakhshnia Y Wu K Govindan and H Soleimani ldquoAnovel hybrid multiple attribute decision-making approach foroutsourcing sustainable reverse logisticsrdquo Journal of CleanerProduction vol 242 Article ID 118461 2020

[23] J J H Liou Y C Chuang E K Zavadskas and G H TzengldquoData-driven hybrid multiple attribute decision-makingmodel for green supplier evaluation and performance im-provementrdquo Journal of Cleaner Production vol 241 ArticleID 118321 2019

[24] Y H Pan and X L Geng ldquoHybrid multiple attribute decisionmaking approach based on Mo-RVIKORrdquo Chinese Journal ofManagement Science vol 27 no 12 pp 143ndash151 2019

[25] M Akram and D Shumaiza ldquoMulti-criteria decision makingbased on q-rung orthopair fuzzy promethee approachrdquo Ira-nian Journal of Fuzzy Systems vol 18 no 5 pp 107ndash127 2021

[26] C Jana G Muhiuddin and M Pal ldquoMulti-criteria decisionmaking approach based on SVTrN Dombi aggregationfunctionsrdquo Artificial Intelligence Review vol 54 no 5pp 3685ndash3723 2021

Mathematical Problems in Engineering 13

[27] C Jana G Muhiuddin and M Pal ldquoMultiple-attribute de-cision making problems based on SVTNH methodsrdquo Journalof Ambient Intelligence and Humanized Computing vol 11no 9 pp 3717ndash3733 2020

[28] C Jana GMuhiuddin andM Pal ldquoSomeDombi aggregationof Q -rung orthopair fuzzy numbers in multiple-attributedecision makingrdquo International Journal of Intelligent Systemsvol 34 no 12 pp 3220ndash3240 2019

[29] G Shahzadi G Muhiuddin M Arif Butt and A AshrafldquoHamacher interactive hybrid weighted averaging operatorsunder fermatean fuzzy numbersrdquo Journal of Mathematicsvol 2021 no 10 17 pages Article ID 5556017 2021

[30] K Zhang J Zhan and X Wang ldquoTOPSIS-WAA methodbased on a covering-based fuzzy rough set An application torating problemrdquo Information Sciences vol 539 pp 397ndash4212020

[31] S F Huang ldquoUsing Linguistic VIKOR and fuzzy cognitivemaps to select virtual reality games development projectrdquoMathematics vol 9 no 11 p 1253 2021

[32] M Akram A Luqman and C Kahraman ldquoHesitant py-thagorean fuzzy ELECTRE-II method for multi-criteria de-cision-making problemsrdquo Applied Soft Computing vol 108Article ID 107479 2021

[33] C Erdin and G Ozkaya ldquoTurkeyrsquos 2023 energy strategies andinvestment opportunities for renewable energy sources Siteselection based on ELECTRErdquo Sustainability vol 11 no 7Article ID 2136 2019

[34] T Wang H Li X Zhou D Liu and B Huang ldquoree-waydecision based on third-generation prospect theory with Z-numbersrdquo Information Sciences vol 569 pp 13ndash38 2021

[35] Z J Du S M Yu H Y Luo and X D Lin ldquoConsensusconvergence in large-group social network environmentcoordination between trust relationship and opinion simi-larityrdquo Knowledge-Based Systems vol 217 Article ID 1068282021

[36] J Wu F Chiclana H Fujita and E Herrera-Viedma ldquoAvisual interaction consensus model for social network groupdecision making with trust propagationrdquo Knowledge-BasedSystems vol 122 pp 39ndash50 2017

[37] L X Chen and N F Luo ldquoPythagorean fuzzy multi-criteriadecision-making based on prospect theoryrdquo Systems Engi-neering Keory amp Practice vol 40 no 3 pp 726ndash735 2020 inChinese

[38] Z S Xu Linguistic Decision Making Springer-Verlag BerlinGermany 2012

[39] R E Moore Methods and Applications of Interval AnalysisPrentice-Hall Hoboken NY USA 1979

[40] R R Ren W W Li M Zhao and X Li ldquoA large groupdecision making method based on public evaluationrdquoManagement Review vol 30 no 10 pp 238ndash247 2018

[41] L Wang Y M Wang and L Martınez ldquoA group decisionmethod based on prospect theory for emergency situationsrdquoInformation Sciences vol 418-419 pp 119ndash135 2017

[42] Y Wu C Xu and T Zhang ldquoEvaluation of renewable powersources using a fuzzy MCDM based on cumulative prospecttheory a case in Chinardquo Energy vol 147 pp 1227ndash1239 2018

[43] M M Salih B B Zaidan A A Zaidan and M A AhmedldquoSurvey on fuzzy TOPSIS state-of-the-art between 2007 and2017rdquo Computers amp Operations Research vol 104 pp 207ndash227 2019

[44] H Xiao Y Zhang X Liu H Yin P Liu and D C Liu ldquoArapid ultrasound vascular disease screening method usingPauTa Criterionrdquo Journal of Physics Conference Seriesvol 1769 Article ID 012009 2021

14 Mathematical Problems in Engineering

[27] C Jana G Muhiuddin and M Pal ldquoMultiple-attribute de-cision making problems based on SVTNH methodsrdquo Journalof Ambient Intelligence and Humanized Computing vol 11no 9 pp 3717ndash3733 2020

[28] C Jana GMuhiuddin andM Pal ldquoSomeDombi aggregationof Q -rung orthopair fuzzy numbers in multiple-attributedecision makingrdquo International Journal of Intelligent Systemsvol 34 no 12 pp 3220ndash3240 2019

[29] G Shahzadi G Muhiuddin M Arif Butt and A AshrafldquoHamacher interactive hybrid weighted averaging operatorsunder fermatean fuzzy numbersrdquo Journal of Mathematicsvol 2021 no 10 17 pages Article ID 5556017 2021

[30] K Zhang J Zhan and X Wang ldquoTOPSIS-WAA methodbased on a covering-based fuzzy rough set An application torating problemrdquo Information Sciences vol 539 pp 397ndash4212020

[31] S F Huang ldquoUsing Linguistic VIKOR and fuzzy cognitivemaps to select virtual reality games development projectrdquoMathematics vol 9 no 11 p 1253 2021

[32] M Akram A Luqman and C Kahraman ldquoHesitant py-thagorean fuzzy ELECTRE-II method for multi-criteria de-cision-making problemsrdquo Applied Soft Computing vol 108Article ID 107479 2021

[33] C Erdin and G Ozkaya ldquoTurkeyrsquos 2023 energy strategies andinvestment opportunities for renewable energy sources Siteselection based on ELECTRErdquo Sustainability vol 11 no 7Article ID 2136 2019

[34] T Wang H Li X Zhou D Liu and B Huang ldquoree-waydecision based on third-generation prospect theory with Z-numbersrdquo Information Sciences vol 569 pp 13ndash38 2021

[35] Z J Du S M Yu H Y Luo and X D Lin ldquoConsensusconvergence in large-group social network environmentcoordination between trust relationship and opinion simi-larityrdquo Knowledge-Based Systems vol 217 Article ID 1068282021

[36] J Wu F Chiclana H Fujita and E Herrera-Viedma ldquoAvisual interaction consensus model for social network groupdecision making with trust propagationrdquo Knowledge-BasedSystems vol 122 pp 39ndash50 2017

[37] L X Chen and N F Luo ldquoPythagorean fuzzy multi-criteriadecision-making based on prospect theoryrdquo Systems Engi-neering Keory amp Practice vol 40 no 3 pp 726ndash735 2020 inChinese

[38] Z S Xu Linguistic Decision Making Springer-Verlag BerlinGermany 2012

[39] R E Moore Methods and Applications of Interval AnalysisPrentice-Hall Hoboken NY USA 1979

[40] R R Ren W W Li M Zhao and X Li ldquoA large groupdecision making method based on public evaluationrdquoManagement Review vol 30 no 10 pp 238ndash247 2018

[41] L Wang Y M Wang and L Martınez ldquoA group decisionmethod based on prospect theory for emergency situationsrdquoInformation Sciences vol 418-419 pp 119ndash135 2017

[42] Y Wu C Xu and T Zhang ldquoEvaluation of renewable powersources using a fuzzy MCDM based on cumulative prospecttheory a case in Chinardquo Energy vol 147 pp 1227ndash1239 2018

[43] M M Salih B B Zaidan A A Zaidan and M A AhmedldquoSurvey on fuzzy TOPSIS state-of-the-art between 2007 and2017rdquo Computers amp Operations Research vol 104 pp 207ndash227 2019

[44] H Xiao Y Zhang X Liu H Yin P Liu and D C Liu ldquoArapid ultrasound vascular disease screening method usingPauTa Criterionrdquo Journal of Physics Conference Seriesvol 1769 Article ID 012009 2021

14 Mathematical Problems in Engineering