Page - 162 - in Algorithms for Scheduling Problems

Algorithms 2018,11, 57 xl isanelementofageneralvectorx(t), λα(t) isadynamicLagrangemultiplierwhichcorresponds to thecomponentsofvectorq (1) α , ρβ(t) isadynamicLagrangemultiplierwhich iscorresponds to thecomponentsofvectorq (2) α , z(o)ijμ isanauxiliaryvariablewhichcharacterizes theexecutionof theoperation α isacurrentnumberofconstraintsq(1)α , β isacurrentnumberofconstraintsq(2)α , ψl(T0) isacomponentofadjointvectorψ(t)at themoment t=T0, ψl(Tf) isacomponentofadjointvectorψ(t)at themoment t=Tf , H(x(t),u(t),ψ(t))=ΨTf(x,u,t) isaHamilton’s function, Φ(ψ(T0)) isan implicit functionofboundarycondition, a isavectorgivenquantityvaluewhichcorresponds to thevectorx(t), Δu isa functionofboundaryconditions, ρ(Tf) isadiscrepancyofboudaryconditions, ψ(r)(T0) isavectorofadjoint systemat themoment, t=T0 r isacurrentnumberof iterationduringscheduleoptimization, Π˜ isaderivativematrix, εu isagivenaccuracyofNewton’smethoditerativeprocedure, Ci isapenaltycoefficient, Δ<i,(r−1)> isacomponentof the functionΔ(r−1)gradientonthe iteration r−1, γ<i,r> isastepofgradient (subgradient)method(algorithm)onthe iteration r˜˜ ψ(0)(T0) isaadjointvectorat themoment t=T0 onthe iteration ′′0′′, ψ˜(r)(T0) isaadjointvectorat themoment t=T0 onthe iteration ′′r′′, σ′ isasomepartof the intervalσ=(T0,Tf ],˜˜N isaassumedoperatorofnewapproximationprocedure, t˜′ <e,(r+1)>, t˜ ′′ <e,(r+1)> are timemomentsofoperation interruption, u∗p(t) isavectorofgeneralizedcontrol inrelaxedproblemofMSOPCconstruction, x∗p(t) isavectorofgeneral state inrelaxedproblemofMSOPCconstruction, ug(t) isavectorofanarbitraryallowablecontrol (allowableschedule), D(i)æ isaoperation“æ”withobject“i”, D(ω)ξ isaoperation“ξ”withobject“ω”, P(i)æ ,P (ω) ξ isacurrentnumberof thebranchandboundmethodsubproblems, J(1)p0 isavalueofscalar formofMSvectorqualitymeasure for thefirst subproblem, J(1)p1 isavalueofscalar formofMSvectorqualitymeasure for thesecondsubproblem, Dx isanattainableset, ϑ isacurrent indexofMScontrolmodel, g isan indexofMSmotioncontrolmodel, k isan indexofMSchannelcontrolmodel, o isan indexofMSoperationcontrolmodel, f isan indexofMSflowcontrolmodel, p isan indexofMSresourcecontrolmodel, e isan indexofMSoperationparameterscontrolmodel, c isan indexofMSstructuredynamiccontrolmodel, ν isan indexofMSauxiliaryoperationcontrolmodel, l isacurrentnumberofMSelementsandsubsystems, u isascalarallowablecontrol input, Θ isacurrentnumberofmodel, æisanumberofoperation“æ”,ξ isanumberofoperation“ξ”,ω isanumberofoperation“ω”, i isacurrentnumberofexternalobject (customer), j isacurrentnumberof internalobject resource, ε1, ε2 areknownconstantswhichcharacterize theaccuracyof iterativesolutionofboundary-valueproblem. δ˜ isastepof integration References 1. Blazewicz, J.;Ecker,K.;Pesch,E.;Schmidt,G.;Weglarz, J.SchedulingComputerandManufacturingProcesses, 2nded.;Springer: Berlin,Germany,2001. 2. Pinedo,M.Scheduling: Theory,Algorithms, andSystems; Springer:NewYork,NY,USA,2008. 162