Resource-constrained project scheduling_ Notation, classific(4)

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6

Table1

BasicnotationSymbol ni

q Yi i3j, iYj red j u j pjRq qkqrjkMjpjmRm mkqrjkm

mrjkm

P.Bruckeretal./EuropeanJournalofOperationalResearch112(1999)3±41

De®nition

setofactivities

numberofrealactivities

setofprecedenceortemporalconstraints

directedgraphofprecedenceortemporalconstraintsprecedenceconstraint

setofdirectpredecessorsofactivityjsetofdirectsuccessorsofactivityjprocessingtimeofactivityjsetofrenewableresources

constantamountofavailableunitsofrenewableresourcekperperiodusageofactivityjofrenewableresourceksetofmodes(processingalternatives)ofactivityjprocessingtimeofactivityjinmodemsetofnonrenewableresources

totalamountofavailableunitsofnonrenewableresourcekperperiodusageofactivityjofrenewableresourcekwhenprocessedinmodem

consumptionofactivityjofnonrenewableresourcekwhenprocessedinmodemstarttimeofactivityjschedule

completiontimeofactivityjvectorofcompletiontimessetoftime-feasibleschedulessetofresource-feasibleschedulessetoffeasibleschedules

resourceconsumptionofresourcekofschedule attimetdeadlineforprojectdurationtimehorizon,indexforperiodsperiods

timeintervalcorrespondingtoperiodt

minimum/maximumtimelagbetweenstartofactivitiesiandj

j

1YFFFY n gj

g g1YFFFYgn STSR

S S S rk Yt "d Yt

t 1Y2YFFFY tÀ1Yt minmax

adijdij

Likewise,for Áandw ÁYÁwewrite andw ,respectively.Examples: mY1Y1 mYI

mresources,1unitofeachresourceavailable,eachactivityrequiresatmost1unitoftheresources

mresources,unlimitednumberofresourceunitsavailable(i.e.,therearenoexplicitresourceconstraints,e.g.inresourceleveling)oneresource

processingtimes,pre generalprecedencecon-straints,etc.pj 1pj sto"dpre

h ins,intree,outtree,treeFFFtemp

allprocessingtimes(activitydurations)areequaltoonestochasticprocessingtimesdeadlineforprojectdurationprecedenceconstraintsbe-tweenactivities

precedencerelationsbetweenactivitiesarespeci®edby

chains,intree,outtree,treeFFFgeneraltemporalconstraintsgivenbyminimumandmaxi-mumstart±starttimelagsbe-tweenactivities

1

b:Activitycharacteristics:Weuseestablishednotationsfrommachinescheduling(cf.[77])likepj