Page - 158 - in Algorithms for Scheduling Problems

Algorithms 2018,11, 57 Theattainabilitysets (AS)ofdynamicmodel (1)havean importantplace inqualitativeanalysisof MScontrolprocesses. Thesesetsallowdetailedanalysisof computationalprocedures to two-point boundaryproblemswithfixedendsof thestate trajectoryx(t)andafixedtimeinterval (T0,Tf ]. AnAS is a fundamental characteristic of anydynamic system. TheASapproachdetermines arangeofexecutionpolicies in thepresenceofdisturbancesoverwhich thesystemcanbeguaranteed tomeetcertaingoals. Theunionof theseexecutionpolicies (i.e., feasible schedules) is calledanAS in the state space. Theunionofpossibleperformanceoutcomes fromthegivenexecutionpolicies is calledanAS in theperformance space [43]. TheAS in the state spacedepicts thepossible states of a schedule subject to variations of the parameters (both planned and perturbation-driven) in thenodes andchannels (e.g., different capacities, lot-sizes, etc.). Let us introduce thenotation for anAS.Dx(t,T0,x(T0),U(x(T0))) is anAS in the state space,DJ(t,T0,x(T0),U(x(T0))) is anAS in the performance indicator’s space, andDξJ(t,T0,x(T0),Ξ,U(x(T0))) is an approximatedASunder disturbancesatmoment t. To interconnect scheduleexecutionandperformanceanalysis to theASin the state space, anASintheperformance spacecanbebrought intocorrespondence (seeFigure4). Table1.Applicationofoptimalcontrolapproaches toscheduling. No ResultsandTheir Implementation TheMainResultsof QualitativeAnalysisof MSControlProcesses TheDirectionsofPractical Implementationof the Results 1 Analysisof solution existence in theproblemsof MScontrol Adequacyanalysisof the controlprocessesdescription in controlmodels 2 Conditionsofcontrollability andattainability in the problemsofMScontrol AnalysisMScontrol technologyrealizabilityonthe planning interval.Detectionof mainfactorsofMSgoaland information-technology abilities. 3 Uniquenessconditionfor optimalprogramcontrols in schedulingproblems Analysisofpossibilityof optimalschedulesobtaining forMSfunctioning 4 Necessaryandsufficient conditionsofoptimality in MScontrolproblems Preliminaryanalysisofoptimal control structure,obtainingof mainexpressions forMS schedulingalgorithms 5 Conditionsof reliabilityand sensitivity inMScontrol problems Evaluationof reliabilityand sensitivityofMScontrol processeswithrespect to perturbation impactsandto thealterationof inputdata contentsandstructure Inprojecting these twoASsontoeachother, a certain rangeof thescheduleexecutionpolicies and the correspondingvariation of theperformance indicators canbedetermined. A continuous timerepresentationallows investigationof the changes inexecutionat each timepoint. Therefore, at each timepoint, anAScanbe calculatedand related tooutputperformance. The justification of the choice of theASmethod is related to its dynamic nature ([43,44]). AnASmay be favourable for obtaining estimations of performance attainability and considerationof theperturbations and attainabilityabilitiesof theschedulesas time functions. In [13,19,33–38],weproposeddifferentmethods andalgorithmsofAScalculations. Theseresultspermit improvementofalgorithmconvergence in MSscheduling. Theproposedmodel interpretsMSschedulingasaresponse toplanninggoalchanges,demand fluctuations,andresourceavailability. In this interpretation, theproblemis toscheduleMSinorder to achieve theplannedgoals (e.g.,MSservice level). Themodel is scalable to othermanagement levels ofMS, i.e., orders and operations can be presentedasMSconfigurationelementsandorders, respectively. Thetransformationofparameters 158