Page - 161 - in Algorithms for Scheduling Problems

Algorithms 2018,11, 57 Withregardto the limitationsof this study, theschedulealgorithmanalysisbyattainablesetswas usedtoanalyse thescheduleoutputsat theendof theplanninghorizon.At thesametime,attainable sets(moreprecisely, theirgeometricapproximations)canalsobeappliedpriortoscheduleoptimization to analyse if any feasible schedule exists for the givenproblemsettings (resource capacities, etc.). This issuehasnotyetbeenconsidered. In lightof therevealedmethodical shortcomingsandapplication limitationsofoptimalcontrol methodsbasedonthemaximumprinciple, the followingfutureresearchavenuescanbestated. First, concreteapplicationcasesneed tobeconsidered forwhichspecific controlmodelsandalgorithms willbedeveloped. Theconstructionofmodelsandcomputationalprocedureswithinprovedaxioms of control theory is important. Second, application of qualitative performance analysismethods forcontrolpolicydynamic investigationunderuncertainty, suchasattainablesets,mustbenamed. These toolsmightbehelpfulwith regard toananalysisofproductionschedule robustness, supply chainresilience,andIndustry4.0systemflexibility. Third, computationalmethods themselves require further investigationandmodificationforconcreteapplicationcases. Therefore,aclosercollaboration between control and industrial engineers is critical for future applications of controlmethods in operationsandfuturesupplychainmanagement. AuthorContributions:B.S.andD.I. conceivedanddesignedtheexperiments;B.S.performedtheexperiments; A.D.andB.S.analyzedthedata;A.D.andD.I. contributedreagents/materials/analysis tools; all authorswrote thepaper. Acknowledgments: The research described in this paper is partially supported by the Russian Foundation for Basic Research (grants 16-07-00779, 16-08-00510, 16-08-01277, 16-29-09482-ofi-i, 17-08-00797, 17-06-00108, 17-01-00139, 17-20-01214, 17-29-07073-ofi-i, 18-07-01272, 18-08-01505), grant 074-U01 (ITMO University), state order of the Ministry of Education and Science of the Russian Federation No. 2.3135.2017/4.6, state research 0073-2018-0003, International project ERASMUS +, Capacity building in higher education, No.73751-EPP-1-2016-1-DE-EPPKA2-CBHE-JP, Innovative teachingandlearningstrategies inopenmodelling andsimulationenvironment forstudents-centeredengineeringeducation. Conflictsof Interest:Theauthorsdeclarenoconflictof interest. Abbreviations AS AttainabilitySets OPC OptimalProgramControl MS ManufacturingSystem SSAM SuccessiveApproximationsMethod AppendixA. ListofNotations x isavectorof theMSgeneral state u isavectorof thegeneralizedcontrol h0,h1 areknownvector functionthatareusedfor thestatexandconditions q(1),q(2) areknownspatio-temporal, technicalandtechnological constraints Jϑ are indicatorscharacterizingMSschedulequality ξ isaknownvector-functionofperturbation influences ψ isavectorofconjugatesystemstate H isaHamilton’s function Job isascalar formof thevectorqualitymeasure Mg isadynamicmodelofMSmotioncontrol, Mk isadynamicmodelofMSchannelcontrol, Mo isadynamicmodelofMSoperationalcontrol, Mf isadynamicmodelofMSflowcontrol, Mp isadynamicmodelofMSresourcecontrol, Me isadynamicmodelofMSoperationparameterscontrol, Mc isadynamicmodelofMSstructuredynamiccontrol, Mν isadynamicmodelofMSauxiliaryoperationcontrol, t isacurrentvalueof time, T0 is start instantof timeof theplanning(scheduling)horizon, Tf isendinstantof timeof theplanning(scheduling)horizon, ψl isacomponentofadjointvectorψ(t), 161