Resource-constrained project scheduling_ Notation, classific(3)

项目进度管理

P.Bruckeretal./EuropeanJournalofOperationalResearch112(1999)3±415

isclassi®cationreadytohaveimportantschemeauni®edfornotationandageneralwhat(cf.isgenerallythatthisschemeprojectiscompatiblescheduling.Itwithisscheduling[77])models(cf.andacceptedinmachinescheduling[18]),resource-constrainedbecausemachinemachinemodels.

arespecialcasesofprojectschedulingschedulingnotation.Inthesequelwewillofplicity,activitiesBasically,(jobs)1weYFFassume®rstproposeFYn.Foraprojectathesaketounifyingofconsistsim-tivityn0inandgeneralauniqueauniquedummydummybeginningac-project 1are(AON)isadded.Frequently,terminationthestructureactivityoftherepresentnetworkdepictedwherebyaso-calledthenodesactivity-on-nodeandthearcstions,precedencerespectively.theactivitiesq and Yi thedenotesprecedencethegraphrela-ofwhilealternativelysingleconstraintssetbyprecedencei3jorconstraints(transitivelyreduced), iYj .aredenoteddirectofdirectofactivitysuccessorspredecessorsofactivitywhilej.The u redprocessing jj isde®nesthesetthetimeofThereisjisagivensetRqbypj.

nonrenewablestrainedand,possibly,ofrenewable,asetofasetRmofspeci®edresources.forrenewableeverynumberperiodofofRenewableunitsmeansdoublythatacon-pre-theplanningofaresourcehorizonisavailable .sourceAsresources,usual,isavailablesaysthatanumberofunitsofNon-are-weskipthefornotiontheentireofdoublyplanningconstrainedhorizon.newablerioddenotedusageandbecauseoftheactivitynonrenewabletheycanbecoveredones.Thebyperthepe-re-q

jofrenewableresourcekis

numberperiod.ofInofbytheunitsrjkwhile qkde®nesthe(constant)

multi-modeofresourcecase,kavailableMineveryjde®nesitymodes,givenj.Thethatis,processingalternativesoftheactiv-setconsumption)bypprocessingThetimeofactivityjinmodemisjm.newable)de®nesresourceofactivityperperiodkisgivenjofusagebyrenewable(totalresourceq(rm

(nonre-jkm)while msourceInjthesingle-modektheavailablenumberforofunitsofrjkmnonrenewablere-k

case,thethatentireisforplanningjMhorizon.superscriptandRm qY,forwejj 1foralltheskipsaketheofmodesimplicity.

indexmandtheof scheduleactivityj(gj)denotescompletionandj.Consequently,thestarttimeg g (completion FY time)1YFFn isa1YFFFYgn isthevectorsibleschedulesschedules,times.Sthesetoftime-fea-ofTde®nesschedules.andSS RSthesetofresource-feasibleR dtisanindexforStimeTthesetoffeasibletimeij

minanddijmax

denoteminimumperiods.andmaximumFinally,itiesbeinteger-valued.

ilags,andjrespectively,.Ingeneral,betweenparametersthearestartassumedofactiv-toalongTable1summarizesthemachineNowwithwesomeextendminornotationintroducedtheadditions.

ajbjc-schemeusedinthetweena:Resourceschedulingenvironment:literature.

Todistinguishbe-projectspeci®cmachineschedulingproblemsand®eldprojectPSscheduling(projectscheduling)problemsweorintroduceinthea-m[18]YrYqaccordingscheduling).tothePSnotationcanbeMPSaugmented(multi-modeofBlaz

ewicztoetPS

al.Innonrenewabletheforcaseresource-constrainedofmulti-modeprojectmachineschedulingscheduling.casew mtheYrYqnotationresourcesYlYsYx.isanalogouslymaybeconsidered.augmentedInalsothisby w project mYrYq

multi-modescheduling

mprojectresourceresources,runitsschedulingofityrequiresavailable,atmosteacheachqunitsactiv-ofw mYrYqYlYsYxthemulti-moderesources

withunitsmrenewableprojectresources,scheduling

able,ofeachresourceavail-rmosteachlqactivityrequiresatunitsnonrenewableunitsoftheresources,able,ofeachresourceresources,smosteachxunitsactivityoftherequiresavail-resourcesatvaluesIfanForofentrytheparametersofmYrYqYlareYsYxspeci®edisreplacedbyÁ,therespectively, mYÁYÁandspeci®edinthefor mYrYÁwewrite minandthe minput.Yr,input,short.weIfallwritevaluesÁinsteadinmYrofYqÁYareÁYÁ.