TOPSIS - Wikipedia

The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) is a multi-criteria decision analysis method, which was originally developed ... TOPSIS FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Thisarticlemaybetootechnicalformostreaderstounderstand.Pleasehelpimproveittomakeitunderstandabletonon-experts,withoutremovingthetechnicaldetails.(April2018)(Learnhowandwhentoremovethistemplatemessage) TheTechniqueforOrderofPreferencebySimilaritytoIdealSolution(TOPSIS)isamulti-criteriadecisionanalysismethod,whichwasoriginallydevelopedbyChing-LaiHwangandYoonin1981[1]withfurtherdevelopmentsbyYoonin1987,[2]andHwang,LaiandLiuin1993.[3] TOPSISisbasedontheconceptthatthechosenalternativeshouldhavetheshortestgeometricdistancefromthepositiveidealsolution(PIS)[4]andthelongestgeometricdistancefromthenegativeidealsolution(NIS).[4] Contents 1Description 2TOPSISmethod 3Normalisation 4Onlinetools 5References Description[edit] Itisamethodofcompensatoryaggregationthatcomparesasetofalternativesbyidentifyingweightsforeachcriterion,normalisingscoresforeachcriterionandcalculatingthegeometricdistancebetweeneachalternativeandtheidealalternative,whichisthebestscoreineachcriterion.AnassumptionofTOPSISisthatthecriteriaaremonotonicallyincreasingordecreasing. Normalisationisusuallyrequiredastheparametersorcriteriaareoftenofincongruousdimensionsinmulti-criteriaproblems.[5][6]CompensatorymethodssuchasTOPSISallowtrade-offsbetweencriteria,whereapoorresultinonecriterioncanbenegatedbyagoodresultinanothercriterion.Thisprovidesamorerealisticformofmodellingthannon-compensatorymethods,whichincludeorexcludealternativesolutionsbasedonhardcut-offs.[7]Anexampleofapplicationonnuclearpowerplantsisprovidedin.[8] TOPSISmethod[edit] TheTOPSISprocessiscarriedoutasfollows: Step1 Createanevaluationmatrixconsistingofmalternativesandncriteria,withtheintersectionofeachalternativeandcriteriagivenas x i j {\displaystylex_{ij}} ,wethereforehaveamatrix ( x i j ) m × n {\displaystyle(x_{ij})_{m\timesn}} . Step2 Thematrix ( x i j ) m × n {\displaystyle(x_{ij})_{m\timesn}} isthennormalisedtoformthematrix R = ( r i j ) m × n {\displaystyleR=(r_{ij})_{m\timesn}} ,usingthenormalisationmethod r i j = x i j ∑ k = 1 m x k j 2 , i = 1 , 2 , … , m , j = 1 , 2 , … , n {\displaystyler_{ij}={\frac{x_{ij}}{\sqrt{\sum_{k=1}^{m}x_{kj}^{2}}}},\quadi=1,2,\ldots,m,\quadj=1,2,\ldots,n} Step3 Calculatetheweightednormaliseddecisionmatrix t i j = r i j ⋅ w j , i = 1 , 2 , … , m , j = 1 , 2 , … , n {\displaystylet_{ij}=r_{ij}\cdotw_{j},\quadi=1,2,\ldots,m,\quadj=1,2,\ldots,n} where w j = W j / ∑ k = 1 n W k , j = 1 , 2 , … , n {\displaystylew_{j}=W_{j}{\Big/}\sum_{k=1}^{n}W_{k},j=1,2,\ldots,n} sothat ∑ i = 1 n w i = 1 {\displaystyle\sum_{i=1}^{n}w_{i}=1} ,and W j {\displaystyleW_{j}} istheoriginalweightgiventotheindicator v j , j = 1 , 2 , … , n . {\displaystylev_{j},\quadj=1,2,\ldots,n.} Step4 Determinetheworstalternative ( A w ) {\displaystyle(A_{w})} andthebestalternative ( A b ) {\displaystyle(A_{b})} : A w = { ⟨ max ( t i j ∣ i = 1 , 2 , … , m ) ∣ j ∈ J − ⟩ , ⟨ min ( t i j ∣ i = 1 , 2 , … , m ) ∣ j ∈ J + ⟩ } ≡ { t w j ∣ j = 1 , 2 , … , n } , {\displaystyleA_{w}=\{\langle\max(t_{ij}\midi=1,2,\ldots,m)\midj\inJ_{-}\rangle,\langle\min(t_{ij}\midi=1,2,\ldots,m)\midj\inJ_{+}\rangle\rbrace\equiv\{t_{wj}\midj=1,2,\ldots,n\rbrace,} A b = { ⟨ min ( t i j ∣ i = 1 , 2 , … , m ) ∣ j ∈ J − ⟩ , ⟨ max ( t i j ∣ i = 1 , 2 , … , m ) ∣ j ∈ J + ⟩ } ≡ { t b j ∣ j = 1 , 2 , … , n } , {\displaystyleA_{b}=\{\langle\min(t_{ij}\midi=1,2,\ldots,m)\midj\inJ_{-}\rangle,\langle\max(t_{ij}\midi=1,2,\ldots,m)\midj\inJ_{+}\rangle\rbrace\equiv\{t_{bj}\midj=1,2,\ldots,n\rbrace,} where, J + = { j = 1 , 2 , … , n ∣ j } {\displaystyleJ_{+}=\{j=1,2,\ldots,n\midj\}} associatedwiththecriteriahavingapositiveimpact,and J − = { j = 1 , 2 , … , n ∣ j } {\displaystyleJ_{-}=\{j=1,2,\ldots,n\midj\}} associatedwiththecriteriahavinganegativeimpact. Step5 CalculatetheL2-distancebetweenthetargetalternative i {\displaystylei} andtheworstcondition A w {\displaystyleA_{w}} d i w = ∑ j = 1 n ( t i j − t w j ) 2 , i = 1 , 2 , … , m , {\displaystyled_{iw}={\sqrt{\sum_{j=1}^{n}(t_{ij}-t_{wj})^{2}}},\quadi=1,2,\ldots,m,} andthedistancebetweenthealternative i {\displaystylei} andthebestcondition A b {\displaystyleA_{b}} d i b = ∑ j = 1 n ( t i j − t b j ) 2 , i = 1 , 2 , … , m {\displaystyled_{ib}={\sqrt{\sum_{j=1}^{n}(t_{ij}-t_{bj})^{2}}},\quadi=1,2,\ldots,m} where d i w {\displaystyled_{iw}} and d i b {\displaystyled_{ib}} areL2-normdistancesfromthetargetalternative i {\displaystylei} totheworstandbestconditions,respectively. Step6 Calculatethesimilaritytotheworstcondition: s i w = d i w / ( d i w + d i b ) , 0 ≤ s i w ≤ 1 , i = 1 , 2 , … , m . {\displaystyles_{iw}=d_{iw}/(d_{iw}+d_{ib}),\quad0\leqs_{iw}\leq1,\quadi=1,2,\ldots,m.} s i w = 1 {\displaystyles_{iw}=1} ifandonlyifthealternativesolutionhasthebestcondition;and s i w = 0 {\displaystyles_{iw}=0} ifandonlyifthealternativesolutionhastheworstcondition. Step7 Rankthealternativesaccordingto s i w ( i = 1 , 2 , … , m ) . {\displaystyles_{iw}\,\,(i=1,2,\ldots,m).} Normalisation[edit] Twomethodsofnormalisationthathavebeenusedtodealwithincongruouscriteriadimensionsarelinearnormalisationandvectornormalisation. LinearnormalisationcanbecalculatedasinStep2oftheTOPSISprocessabove.VectornormalisationwasincorporatedwiththeoriginaldevelopmentoftheTOPSISmethod,[1]andiscalculatedusingthefollowingformula: r i j = x i j ∑ k = 1 m x k j 2 , i = 1 , 2 , … , m , j = 1 , 2 , … , n {\displaystyler_{ij}={\frac{x_{ij}}{\sqrt{\sum_{k=1}^{m}x_{kj}^{2}}}},\quadi=1,2,\ldots,m,\quadj=1,2,\ldots,n} Inusingvectornormalisation,thenon-lineardistancesbetweensingledimensionscoresandratiosshouldproducesmoothertrade-offs.[9] Onlinetools[edit] DecisionRadar :AfreeonlineTOPSIScalculatorwritteninPython. Yadav,Vinay;Karmakar,Subhankar;Kalbar,PradipP.;Dikshit,A.K.(January2019)."PyTOPS:APythonbasedtoolforTOPSIS".SoftwareX.9:217–222.doi:10.1016/j.softx.2019.02.004. References[edit] ^abHwang,C.L.;Yoon,K.(1981).MultipleAttributeDecisionMaking:MethodsandApplications.NewYork:Springer-Verlag. ^Yoon,K.(1987)."Areconciliationamongdiscretecompromisesituations".JournaloftheOperationalResearchSociety.38(3):277–286.doi:10.1057/jors.1987.44. ^Hwang,C.L.;Lai,Y.J.;Liu,T.Y.(1993)."Anewapproachformultipleobjectivedecisionmaking".ComputersandOperationalResearch.20(8):889–899.doi:10.1016/0305-0548(93)90109-v. ^abAssari,A.,Mahesh,T.,&Assari,E.(2012b).Roleofpublicparticipationinsustainabilityofhistoricalcity:usageofTOPSISmethod.IndianJournalofScienceandTechnology,5(3),2289-2294. ^Yoon,K.P.;Hwang,C.(1995).MultipleAttributeDecisionMaking:AnIntroduction.California:SAGEpublications. ^Zavadskas,E.K.;Zakarevicius,A.;Antucheviciene,J.(2006)."EvaluationofRankingAccuracyinMulti-CriteriaDecisions".Informatica.17(4):601–618.doi:10.15388/Informatica.2006.158. ^Greene,R.;Devillers,R.;Luther,J.E.;Eddy,B.G.(2011)."GIS-basedmulti-criteriaanalysis".GeographyCompass.5/6(6):412–432.doi:10.1111/j.1749-8198.2011.00431.x. ^Locatelli,Giorgio;Mancini,Mauro(2012-09-01)."Aframeworkfortheselectionoftherightnuclearpowerplant"(PDF).InternationalJournalofProductionResearch.50(17):4753–4766.doi:10.1080/00207543.2012.657965.ISSN 0020-7543. ^Huang,I.B.;Keisler,J.;Linkov,I.(2011)."Multi-criteriadecisionanalysisinenvironmentalscience:tenyearsofapplicationsandtrends".ScienceoftheTotalEnvironment.409(19):3578–3594.doi:10.1016/j.scitotenv.2011.06.022.PMID 21764422. 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