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CORRESPONDENCETOSKRAVANJA,FACULTYOFCIVILENGINEERING,UNIVERSITYOFMARIBOR,SLOVENIA17,SI2000MARIBOR,SLOVENIAEMAILSTOJANKRAVANJAUNIMBSICCC00295981/98/020329361750RECEIVED12AUGUST19961998JOHNWILEYOPTIMIZATIONTOPOLOGYOPTIMIZATIONDISCRETEVARIABLEOPTIMIZATIONMIXEDINTEGERNONLINEARPROGRAMMINGMINLPTHEMODIEDOA/ERALGORITHMMINLPSTRATEGYHYDRAULICGATESLIDINGGATEROLLERGATEASWAN1INTRODUCTIONTHISPAPERDESCRIBESTHEMIXEDINTEGERNONLINEARPROGRAMMINGMINLPAPPROACHTOTHESYNTHESISOFROLLERANDSLIDINGGATESTRUCTURES,IETHESIMPLESTTYPESAMONGVERTICALLIFTHYDRAULICSTEELGATES,SEEFIGURE1ROLLERANDSLIDINGGATESAREALSOREGARDEDASTHEMOSTFREQUENTLYFIGURE1VERTICALLIFTHYDRAULICSTEELGATESTRUCTUREMANUFACTUREDTYPESOFHYDRAULICSTEELGATESFORHEADWATERCONTROLTHEYAREUSEDTOREGULATETHEWATERSTREAMONHYDROELECTRICPLANTS,DAMSORSPILLWAYSASHYDRAULICSTEELGATESAREVERYSPECIALSTRUCTURES,ONLYAFEWAUTHORSHAVEDISCUSSEDTHEIROPTIMIZATION,EGKRAVANJAETAL,13JONGELINGANDKOLKMAN4ASWELLASALMQUISTETAL5PARTICULARINTERESTWASSHOWNINTHEOPTIMIZATIONNOTOFTHESEROLLERANDSLIDINGGATESBUTOFSIMILARSTRUCTURESINSUCHINVESTIGATIONS,VANDERPLAATSANDWEISSHAAR6ASWELLASGURDALETAL7OPTIMIZEDSTIENEDLAMINATEDCOMPOSITEPANELS,BUTLER8ANDRINGERTZ9OPTIMIZEDSTIENEDPANELS,FARKASANDJARMAI10OPTIMIZEDWELDEDRECTANGULARCELLULARPLATES,FINCKENORETAL11TREATEDSKINSTRINGERCYLINDERSANDGENDYETAL12STIENEDPLATESALMOSTALLAUTHORSUSEDNONLINEARPROGRAMMINGNLPTECHNIQUESGURDALETAL7PROPOSEDTHEGENETICALGORITHM,WHILEKRAVANJAETAL13INTRODUCEDMINLPALGORITHMSANDSTRATEGIESTOTHESIMULTANEOUSTOPOLOGYANDPARAMETEROPTIMIZATIONOFTHEGATEINPARTSIOFTHISTHREEPARTSERIESOFPAPERS,AGENERALVIEWOFTHEMINLPAPPROACHTOTHESIMULTANEOUSTOPOLOGYANDPARAMETEROPTIMIZATIONOFSTRUCTURESISPRESENTEDPARTIIDESCRIBESTHEEXTENSIONTOTHESIMULTANEOUSSTANDARDDIMENSIONOPTIMIZATIONBASEDONTHESUPERSTRUCTUREAPPROACHDENEDINPARTSIANDII,THEMAINOBJECTIVEOFTHISPAPERPARTIIIISTHEMINLPSYNTHESISOFROLLERANDSLIDINGHYDRAULICSTEELGATESTRUCTURES,OBTAINEDATMINIMALGATECOSTSANDSUBJECTEDTODENEDDESIGN,MATERIAL,STRESS,DEECTIONANDSTABILITYCONSTRAINTSASTHEMINLPAPPROACHENABLESSIMULTANEOUSTOPOLOGY,PARAMETERANDSTANDARDDIMENSIONOPTIMIZATION,ANUMBEROFGATESTRUCTURALELEMENTSGIRDERSANDPLATES,THEGATEGLOBALGEOMETRY,INTERMEDIATEDISTANCESBETWEENSTRUCTURALELEMENTSANDALLCONTINUOUSANDSTANDARDDIMENSIONSAREOBTAINEDSIMULTANEOUSLYTHISLASTPARTOFTHETHREEPARTSERIESOFPAPERSISDIVIDEDINTOTHREEMAINSECTIONS1SECTION2DESCRIBESHOWDIERENTTOPOLOGYANDSTANDARDDIMENSIONALTERNATIVESAREPOSTULATEDANDHOWTHEIRINTERCONNECTIONRELATIONSAREFORMULATEDBYMEANSOFEXPLICITLOGICALCONSTRAINTSINORDERTOPERFORMTOPOLOGYANDSTANDARDDIMENSIONALTERATIONSWITHINTHEOPTIMIZATIONPROCEDURE330SKRAVANJA,ZKRAVANJAANDBSBEDENIKINTJNUMERMETHENGNG43,32936419981998JOHNWILEYBMINIMALSTRUCTUREWITHONLYXEDSTRUCTURALELEMENTSGIRDERSASTHEMODELLINGOFVERTICALGIRDERSISSIMPLIEDANDNEEDSNOSPECIALINTERCONNECTIONLOGICALCONSTRAINTS,THEMODELLINGOFDISCRETEDECISIONSREGARDINGHORIZONTALGIRDERSPROVEDTOBEMORESOPHISTICATED221MODELLINGOFTOPOLOGYALTERATIONSLETUSCONSIDERTHEVERTICALCROSSSECTIONOFTHEGATEELEMENTSUPERSTRUCTUREWITHXEDANDALTERNATIVEHORIZONTALGIRDERS,SEEFIGURE3ATHENUMBEROFXEDANDALTERNATIVEGIRDERSANDTHEIRLOCATIONSINTHESUPERSTRUCTURECANBEDESCRIBEDBYTHEFOLLOWINGLOGICALCONSTRAINTSM/M|MYMM99YMYM1,M“2,3,M110YM9“111LOGICALCONSTRAINT9DENESTHEMINIMALM/ANDMAXIMALM9NUMBEROFSTRUCTURALELEMENTSGIRDERSWHILENUMBERM/REPRESENTSTHENUMBEROFXEDSTRUCTURALELEMENTS,THEDIERENCEBETWEENTHEMAXIMALANDMINIMALNUMBEROFELEMENTSM9M/GIVESTHENUMBEROFALTERNATIVESTRUCTURALELEMENTSCONSTRAINT10DENESTHEDIRECTIONOFTHEREMOVALOFALTERNATIVEELEMENTSFROMTHETOPDOWNTHESUPERSTRUCTUREFROMFIGURE3ISEVIDENTTHATTHEMOSTUPPERELEMENTISTHEXEDONE,WHICHISSETBYTHECONSTRAINT11ITTHENFURTHERFOLLOWSFROM334SKRAVANJA,ZKRAVANJAANDBSBEDENIKINTJNUMERMETHENGNG43,32936419981998JOHNWILEYHERETHEYAREPROPOSEDTOBECALCULATEDBYCONSTRAINTS21THENEXTCONSTRAINTS2428EXPRESSNORMALANDSHEARSTRESSES,ACTINGINTHESECTIONOFHORIZONTALGIRDERSTHECOMBINATIONSBETWEENTHESESTRESSESARENOTDECISIVE,BECAUSEP9ACTSINTHEMIDDLEOFTHEGIRDERANDQ9ACTSATBOTHGIRDERENDSCONSTRAINTS2931EXPLAINTHELOCALBUCKLINGCONSTRAINTSFORTHEPLATESECTIONSOFGIRDERWEBSCONSTRAINTS29AREACTIVATEDWHENTHEWATERLOADACTSONTHEOUTERSIDEOFTHESKINPLATEANDCONSTRAINTS30AREACTUATEDWHENWATERACTSONTHEINNERSIDEOFTHESKINPLATETHEPLATEBUCKLINGREDUCTIONCOECIENTS,DEPENDENTONTHEPLATESLENDERNESSRATIOJP,ISDENEDBYEUROCODE317S“06/JJ2P013INTHISPAPER,THEBUCKLINGCOECIENTSISDETERMINEDBYTHEEQUATIONS“1/022703175365J1LATERALBUCKLINGOFHORIZONTALGIRDERSISVERIEDBYCONSTRAINTS32INTHECASEOFCOMPRESSEDGIRDERANGESWHENWATERACTSONTHEINNERSIDEOFTHESKINPLATEALLSTABILITYCONSTRAINTS2932AREDENEDACCORDINGTOEUROCODE3THEHORIZONTALGIRDERDEECTIONISEXPRESSEDBYMEANSOFTHEINEQUALITYCONSTRAINT33ALLCONSTRAINTSOFTHELOWERANDUPPERHORIZONTALGIRDERSAREDENEDINTHESAMEMANNERASINTERMEDIATEGIRDERS,WITHTHEEXCEPTIONTHATDIERENTCOECIENTSOFTHEEECTIVEWIDTHLN,MAREUSEDHERE,SEECONSTRAINTS34AND35THEMINLPOPTIMIZATIONAPPROACHTOSTRUCTURALSYNTHESISPARTIII3391998JOHNWILEYM“2,3,M1FORTHEGIRDER,M“1,2,M2FORTHELOWERANDM“2,3,M1FORTHEUPPERSKINPLATEELEMENTP213N,M1P1N,MKP13N,M12P13N,M1P1N,MKP13N,M1P23M“MFORTHEGIRDERANDM“M1FORTHESKINPLATEELEMENTP213N,M1P1N,MKP13N,M12P13N,M1P1N,MKP13N,M1P23M“1,2,M170DHSN“HN,MHSN,N“1,2,N71HN,MHSNL4/DG,N“1,2,N72HN,MHN,M1L4/DG,N“1,2,NM“2,3,M173HWN/1“H8,74HWN“HWN1H0N1,N“2,3,N75H505“N|NH0N76HGN“H0NHGNHSN,N“1,2,N77HN,M“H0NHGN,N“1,2,N78TFNBWNTSN“TFN1BWN1TSN1,N“2,3,N7934PUREINTEGERLOGICALCONSTRAINTSTHESETOFCONSTRAINTS8082REPRESENTSINTEGERLOGICALLINEARCONSTRAINTSCONSTRAINTS80DETERMINETHENUMBEROFXEDSTRUCTURALELEMENTSXEDHORIZONTALGIRDERSANDSKINPLATEELEMENTSFOREACHGATEELEMENTCONSTRAINTS81MAKESURETHATTHENUMBEROFHORIZONTALGIRDERSANDSKINPLATEELEMENTSOFTHEUPPERGATEELEMENTSISEQUALTOORHIGHERTHANTHEONEOFTHELOWERGATEELEMENTSCONSTRAINTS82EXPLAINTHATTHEUPPERALTERNATIVEINTERMEDIATEHORIZONTALGIRDERSANDSKINPLATEELEMENTSDISAPPEARINTHEDIRECTIONFROMTHETOPTOTHEBOTTOMOFTHEGATEBEFORETHELOWERONESDOTHIS,INEECT,REDUCESTHECOMBINATORICSOFTHEMINLPPROBLEMSINCEITTHEMINLPOPTIMIZATIONAPPROACHTOSTRUCTURALSYNTHESISPARTIII3451998JOHNWILEYM“2,3,M18235INTERCONNECTIONLOGICALCONSTRAINTSINTERCONNECTIONLOGICALMIXEDINTEGERLINEARCONSTRAINTSSTARTWITHCONSTRAINT83,WHICHDENESTHEGATEPARTITIONINTHEHORIZONTALDIRECTIONCONSTRAINTS8487DENEVERTICALDISTANCEHSNBETWEENTHEUPPERMOSTSELECTEDINTERMEDIATEHORIZONTALGIRDERANDTHESILLOFTHEGATEELEMENTCONSEQUENTLY,THEUPPERMOSTXEDHORIZONTALGIRDERISALWAYSCONNECTEDTOTHEUPPERMOSTSELECTEDINTERMEDIATEALTERNATIVEHORIZONTALGIRDERANDDISCONNECTEDFROMTHEDISAPPEARINGONESCONSTRAINTS8891DETERMINETHEMINIMALOPENINGSBETWEENANGESOFHORIZONTALGIRDERSDG“22V|VYN,V,N“183HSNHN,MH614/1YN,M1H614/1YN,MH614/YN,M1,N“1,2,NM“2,M284HSNHN,MH614/1YN,M1H614/1YN,MH614/YN,M1,N“1,2,NM“2,M285HSNHN,M1H614/1YN,M1,N“1,2,N86HSNHN,M1H614/1YN,M1,N“1,2,N87HN,2BFN,2/2H08/BF08NHM“3,4,M189HN,MBFUPNHN,MBFN,M/2HM“1,2,M190HN,MBFUPNHN,M1BFN,M1/2HIFNOT,THEBOUNDSARESETTOZEROBFN,MB61FN,MYN,M,N“1,2,NM“1,2,M92BFN,MB08,9FN,MYN,M,N“1,2,NM“1,2,M93TWN,MT61WN,MYN,M,N“1,2,NM“1,2,M94TWN,MT08,9WN,MYN,M,N“1,2,NM“1,2,M95DHN,MD61HN,MYN,M,N“1,2,NM“1,2,M196DHN,MD08,9HN,MYN,M,N“1,2,NM“1,2,M197BWIN,MB61WNYN,M,N“1,2,NM“1,2,M98BWIN,MB08WNYN,M,N“1,2,NM“1,2,M99TFIN,MT61FNYN,M,N“1,2,NM“1,2,M100TFIN,MT08FNYN,M,N“1,2,NM“1,2,M101362OGICALRELATIONSFORCOMMONVARIABLESMIXEDINTEGERLINEARCONSTRAINTS102105DENETHELOGICALRELATIONSOFTHECOMMONDESIGNVARIABLESTHESECONSTRAINTSENFORCEUPONTHECOMMONDESIGNVARIABLESOFTHEHORIZONTALGIRDERSBWIN,MANDTFIN,MTHEVALUESOFTHECORRESPONDINGCOMMONVARIABLESOFTHEENTIREGATEBWNANDTFNBWIN,MBWNB61WN1YN,M,N“1,2,NM“1,2,M102BWIN,MBWNB61WN1YN,M,N“1,2,NM“1,2,M103TFIN,MTFNT61FN1YN,M,N“1,2,NM“1,2,M104TFN,MTFNT61FN1YN,M,N“1,2,NM“1,2,M10537OGICALCONSTRAINTSFORSTANDARDVARIABLES371STANDARDDIMENSIONLOGICALCONSTRAINTSSTANDARDDIMENSIONLOGICALCONSTRAINTSDENEFOURSTANDARDDIMENSIONSTHICKNESSOFTHESKINPLATETSN,ANGETHICKNESSOFTHEHORIZONTALGIRDERTFN,THEWEBTHICKNESSOFTHEOUTERHORIZONTALGIRDERT065WN,MANDTHEWEBTHICKNESSOFTHEINNERHORIZONTALGIRDERT/WN,MEACHSTANDARDDIMENSIONISDETERMINEDBYASCALARPRODUCTBETWEENITSCORRESPONDINGVECTOROFBINARYVARIABLESYANDITSASSIGNEDVECTOROFDISCRETECONSTANTSQCONSTRAINTS106,108,110,111,113AND114,WHEREONLYONEDISCRETEVALUECANBESELECTEDY“1,SEECONSTRAINTS107,109,112AND115NOTETHATD03DENOTESCORROSIONADDITIONTOTHESHEETIRONPLATESINTHESYNTHESIS,THESEDIMENSIONSARERSTTREATEDASCONTINUOUSANDLATERASSTANDARDDIMENSIONSTHEBOUNDLOGICALCONSTRAINTS94AND95FORTHEDESIGNVARIABLETWN,MARETHEREFORESUITABLEFORTHESTANDARDDIMENSIONST065WN,MANDT/WN,MASSTANDARDDIMENSIONSTSNANDTFNCORRESPONDTOTHECOMMONSTANDARDDESIGNVARIABLESFORTHEMINLPOPTIMIZATIONAPPROACHTOSTRUCTURALSYNTHESISPARTIII3471998JOHNWILEYM“1ORM“M110T065WN,MP|PQ/,1D03YN,PT065,UPWN,M1YN,N“1,2,NM“1ORM“M111P|PYN,P“1,N“1,2,N112T/WN,MR|RQ/,3D03YN,RT/,UPWN,M1YN,N“1,2,NM“1,2,M113T/WN,MR|RQ/,3D03YN,RT/,UPWN,M1YN,N“1,2,NM“1,2,M114R|RYN,R“1N“1,2,N11538CONSTRAINTSFORTHEMILPPHASEONLYADDITIONALMILPCONSTRAINTS116121,CONSIDEREDONLYINTHEMILPPHASEOFTHEMINLPOPTIMIZATION,AREINCLUDEDINTHEMODELINORDERTOPERFORMTHEOPTIMIZATIONFASTERANDMOREECIENTLYTOENFORCEFASTERMINLPCONVERGENCE,EQUALVERTICALDISTANCESBETWEENTHESELECTEDHORIZONTALGIRDERSAREPROPOSEDDHN,MHN,MH08/M91D61/,1YN,M1,N“1,2,NM“1,2,M1116DHN,MHN,MH08/M91D61/,1YN,M1,N“1,2,NM“1,2,M1117DHN,MHN,MH08/M92D61/,YN,M1D61/,1YN,M2,N“1,2,NM“1,2,M2118348SKRAVANJA,ZKRAVANJAANDBSBEDENIKINTJNUMERMETHENGNG43,32936419981998JOHNWILEYM“1,2,M2119DHN,MHN,MH08/M93D61/,YN,M2N“1,2,NM“1,2,M3120DHN,MHN,MH08/M93D61/,YN,M2N“1,2,NM“1,2,M312139HEURISTICCONSTRAINTSHEURISTICCONSTRAINTSADDITIONALLYINCREASETHEECIENCYOFTHEMINLPOPTIMIZATIONANDAREADDEDONLYFORTHEMILPPHASECONSTRAINTS122125SHOWTHATTHETHICKNESSESOFEACHUPPERGATEELEMENTMUSTNOTEXCEEDTHEONESOFTHELOWERGATEELEMENTSEACHUPPERMAINGATEELEMENTISHIGHERTHANITSADJOININGLOWERMAINGATEELEMENTS,SEECONSTRAINTS126TSNTSN1,N“2,3,N122TFNTFN1,N“2,3,N123TWN,MTWN1,M,N“2,3,NM“1ORM“M124TWN,MTWN1,M/2,N“2,3,NM“2,3,M1125HONHON1N“2,3,N126310INITIALIZATIONOFVARIABLESTHEINITIALIZATIONOFCONTINUOUSANDBINARYVARIABLESISALSOINCLUDED,SEELISTOFSYMBOLSINAPPENDIXIONLYDESIGNVARIABLESHAVETOBEINITIALISED,WHILENONDESIGNVARIABLESARECALCULATEDONTHEBASISOFTHEINITIALIZEDDESIGNVARIABLES4SYNTHESISOFTHEINTAKEGATEINASWANACOMPREHENSIVECOMPARATIVEDESIGNRESEARCHWORKHASBEENP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MANN20,21TOPISTHEIMPLEMENTATIONOFMANYADVANCEDOPTIMIZATIONTECHNIQUES,AMOSTIMPORTANTOFWHICHISTHEMODIEDOA/ERALGORITHMMINOSBYMURTAGHANDSAUNDERS22ISUSEDTOSOLVENLPSUBPROBLEMSANDOSL23ISUSEDTOSOLVEMILPMASTERPROBLEMSINTERMSOFCOMPLEXITY,THETOPSSYNTHESISPROBLEMSCANRANGEFROMASIMPLENLPOPTIMIZATIONPROBLEMOFASINGLESTRUCTUREUPTOTHEMINLPOPTIMIZATIONOFACOMPLEXSUPERSTRUCTUREPROBLEMTOPRUNSAUTOMATICALLYORINANINTERACTIVEMODEANDTHUSPROVIDESTHEUSERWITHAGOODCONTROLANDSUPERVISIONOFTHECALCULATIONSTHEOPTIMALSOLUTIONYIELDSTHECOSTSOF49387OR300PERCENTOFNETSAVINGSTHEOPTIMUMWASREACHEDATTHETOPOLOGYOF455/54HORIZONTALGIRDERSFORTHELOWER,5FORTHEINTERMEDIATEAND5FORTHEUPPERMAINGATEELEMENTASWELLAS5VERTICALGIRDERSFORTHEENTIREGATEWITHTHECORRESPONDINGNUMBEROFSKINPLATEELEMENTSTHEMODIEDOA/ERALGORITHMCONVERGEDCONSIDERABLYFASTONLY4MAJORMINLPITERATIONSWERENEEDED3MILPAND4NLPSUBPROBLEMS,SEETABLESVIANDVIIITTOOKONLY161SOFCPUONTHECOMPUTERVAX4600TOOBTAINTHEOPTIMALRESULTBESIDETHEOPTIMALTOPOLOGYOF455/5OFTHEGATESTRUCTURE,THEOPTIMIZATIONALSOGAVEANOPTIMALGATEGEOMETRYANDALLTHESIZESFIGURE8SHOWSTHEINITIALANDTHREEOTHERTHEMINLPOPTIMIZATIONAPPROACHTOSTRUCTURALSYNTHESISPARTIII3531998JOHNWILEYBOBTAINEDTOPOLOGY445/5,2ITERATIONCOBTAINEDTOPOLOGY555/5,3ITERATIONDOBTAINEDOPTIMALTOPOLOGY455/5,4ITERATIONTOPOLOGIESTHELASTONEISTHEOPTIMAL,AUTOMATICALLYOBTAINEDANDSEQUENTIALLYCALCULATEDATEACHMINLPITERATIONITSHOULDBENOTEDTHATTHEOBTAINEDRESULTOF49387HASADRAWBACKINTHATITISREACHEDONTHEASSUMPTIONTHATALLDIMENSIONS,INCLUDINGTHETHICKNESSESOFSHEETIRONPLATES,ARECONTINUOUSSTANDARDDIMENSIONSARETHUSNOTSIMULTANEOUSLYDEALTWITHINTHEOPTIMIZATIONWECALLSUCHRESULTSTHEORETICALOPTIMAHOWEVER,TOUPGRADETHEPRACTICALRELEVANCEOFTHEOBTAINEDRESULTS,STANDARDDIMENSIONSMUSTALSOBECONSIDEREDWECALLSUCHPRACTICALRESULTSREALOPTIMATHESIMPLESTWAYOFACCOMPLISHINGTHISTASKISTOROUNDUPTHETHICKNESSESTOTHEIRCLOSESTANDARDDIMENSIONVALUESTOOBTAINAFEASIBLERESULT,THEROUNDINGUPHASTOBEDIRECTEDTOTHECLOSESTHIGHERSTANDARDVALUESTHEFOLLOWINGNLPOPTIMIZATION,PERFORMEDATTHEXEDOPTIMALTOPOLOGYOF455/5,THUSINVOLVESALLTHESTANDARDDIMENSIONSSETTOTHEAPPROPRIATEVALUES,WHILEOTHERDIMENSIONSAREVARIEDITSHOULDBENOTEDTHATTHISSEQUENTIALOPTIMIZATIONOFSTANDARDDIMENSIONSMAYLEADTOSUBOPTIMALRESULTSANDTHATSPECIALCARESHOULDBETAKENTOAVOIDINFEASIBLERESULTSWHENTHEDIMENSIONSAREROUNDEDUPTOTHEIRSTANDARDVALUESITWILLBESEENINTHEFOLLOWINGSUBCHAPTERTHAT,INORDERTOAVOIDTHEDRAWBACKS,ALLSTANDARDDIMENSIONSSHOULDBEOPTIMIZEDSIMULTANEOUSLY,TOGETHERWITHTHETOPOLOGYANDCONTINUOUSPARAMETERSOFTHEGATEFIGURE9SHOWSTHEINTAKEGATEWITHTHEROUNDEDUPTHICKNESSESOFSHEETIRONPLATESANDOTHEROPTIMALDIMENSIONSTHISREALGATEOPTIMUMYIELDSTHECOSTSOF50578OR283PERCENTOFNETSAVINGS354SKRAVANJA,ZKRAVANJAANDBSBEDENIKINTJNUMERMETHENGNG43,32936419981998JOHNWILEYBOPTIMALTOPOLOGY455/5,294OFNETSAVINGSIMPLEMENTATIONOFTHEMODIEDOA/ERALGORITHMTOSOLVEHIGHLYCOMBINATORIAL,NONLINEARANDNONCONVEXPROBLEMS,THELINKEDTWOPHASEMINLPSTRATEGYWASINTRODUCEDINORDERTOACCELERATETHECONVERGENCEOFTHE