Page - 148 - in Algorithms for Scheduling Problems

Algorithms 2018,11, 57 Mν—dynamicmodelofMSauxiliaryoperationcontrol. Thedetailedmathematical formulationof thesemodelswaspresentedby[36,37]aswellas [38]. Weprovide thegeneralizeddynamicmodelofMScontrolprocesses (Mmodel)below: M= { u(t) ∣∣ .x= f(x,u,t);h0(x(T0))≤O, h1(x(Tf))≤O,q(1)(x,u)=O,q(2)(x,u)<O } , (1) Jϑ= Jϑ(x(t),u(t),t)=ϕϑ ( x(tf) ) + Tf∫ T0 fϑ(x(τ),u(τ),τ)dτ, ϑ∈{g, k, o, f, p, e, c,ν}, where ϑ ∈ {g, k, o, f, p, e, c,ν}—lower index which correspond to the motion control model, channel controlmodel; operations controlmodel; flow control; Mp—resource control; operation parameters control; structure dynamic control model; auxiliary operation control model; x = ‖x(g)T,x(k)T,x(o)T,x(p)T,x(f)T,x(e)T,x(c)T,x(v)T‖T is a vector of the MS generalized state, u = ‖u(g)T,u(k)T,u(o)T,u(p)T,u(f)T,u(e)T,u(c)T,u(v)T‖T isavectorofgeneralizedcontrol,h0,h1 areknown vectorfunctionsthatareusedforthestatexendconditionsatthetimepoints t=T0and t=Tf ,andthe vector functionsq(1),q(2)definethemainspatio–temporal, technical, andtechnological conditions for MSexecution; Jϑ are indicatorscharacterizingthedifferentaspectsofMSschedulequality. Overall, the constructed model M (1) is a deterministic, non-linear, non-stationary, finite-dimensional differential system with a reconfigurable structure. Figure 1 shows the interconnectionofmodelsMg,Mk,Mo,Mp,Mf ,Me,Mc, andMν embeddedin thegeneralizedmodel. InFigure1 theadditionalvector functionξ=‖ξ(g)T,ξ(k)T,ξ(o)T,ξ(p)T,ξ(f)T,ξ(e)T,ξ(c)T,ξ(v)T‖T ofperturbation influences is introduced. The solutionsobtained in thepresentedmulti-model complex are coordinatedby the control inputsvectoru(o)(t)of themodelMo.Thisvectordetermines thesequenceof interactionoperations and fixes the MS resource allocation. The applied procedure of solution adjustment refers to resourcecoordination. The model complex M evolves and generalizes the dynamic models of scheduling theory. Thepredominantdistinctive featureof thecomplex is thatnon-linear technological constraintsare actualizedintheconvexdomainofallowablecontrol inputsratherthanindifferentialequations[36,37]. )(QuMk 1 Mp 5 Mg 2 Mo 3 MQ 4 Mf 6 Me 7 Mc 8 )( 0 kx )(kξ )(ku )(kx )(gx )( 0 gx )(gξ )( 0 ku )( 0 ox )(oξ )(pu )( 0 Qx )(Qξ )(kJ )(oJ )(QJ )(Qx )( 0 px )(px )(gu )( fu )( 0 fx )(ox )()( fT Qx )( fJ )( fξ )( fx )( 0 ex )( )( f e Tx )(eu )(ñu )(cx )( 0 cx )(cξ )()()( )()()( ,, ,, efp ogk JJJ JJJ )(ñJ )(eξ )(kJ )(ex Figure1.Theschemeofoptimalprogramcontrolmodel interconnection. 148