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Algorithms 2018,11, 57
dimensionality, convexity, etc. In general, optimal control algorithms only provide the necessary
conditions for optimal solution existence. Themaximumprinciple provides both optimality and
necessary conditions only for some specific cases, i.e., linear control systems. As such, further
investigationsarerequired ineachconcreteapplicationcase.
Thispaperseeks tobring thediscussionforwardbycarefullyelaboratingontheoptimality issues
describedaboveandprovidingsomeideasandimplementationguidanceonhowtoconfront these
challenges. Thepurposeof thepresent study is tocontribute toexistingworksbyprovidingsome
closedformsofalgorithmicoptimalityprovenin therichoptimalcontrolaxiomatic.
The rest of this paper is organized as follows. In Section 2,wepropose general elements of
themultiple-modeldescriptionof industrialproductionschedulinganditsdynamic interpretation.
InSection3, optimal control computational algorithmsand their analyses areproposed. Section4
presentsacombinedmethodandalgorithmforshort-termscheduling inMS. InSection5,qualitative
andquantitativeanalysisof theMSschedulingproblemissuggested. Thepaperconcludes inSection6
bysummarizingtheresultsof this study.
2.OptimalControlApplications toSchedulinginManufacturingSystems
Optimalcontrolproblemsbelongto theclassofextremumoptimizationtheory, i.e., theanalysis
of theminimizationormaximizationofsomefunctions ([21–28]. This theoryevolvedonthebasisof
calculusvariationprinciplesdevelopedbyFermat,Lagrange,Bernulli,Newton,Hamilton,andJacobi.
Inthe20thcentury, twocomputationalfundamentalsofoptimalcontroltheory,maximumprinciple[21]
anddynamicprogrammingmethod[29],weredeveloped. Thesemethodsextendtheclassical calculus
variationtheorythat isbasedoncontrolvariationsofacontinuous trajectoryandtheobservationof
theperformance impactof thesevariationsat the trajectory’s terminus. Sincecontrol systemsinthe
middleof the20thcenturywereincreasinglycharacterizedbycontinuousfunctions(suchas0–1switch
automats), thedevelopmentofboththemaximumprincipleanddynamicprogrammingwasnecessary
inorder tosolve theproblemwithcomplexconstraintsoncontrolvariables.
Manufacturingmanagersarealways interested innon-deterministicapproaches toscheduling,
particularlywherescheduling is interconnectedwith thecontrol function [30]. Studiesby[31]and[11]
demonstrated awide range of advantages regarding the application of control theoreticmodels
in combinationwith other techniques for production and logistics. Among others, they include
anon-stationaryprocessviewandaccuracyofcontinuous time. Inaddition,awiderangeofanalysis
tools fromcontrol theoryregardingstability, controllability,adaptability, etc.maybeusedifaschedule
is described in terms of control. However, the calculation of the optimal program control (OPC)
with thedirectmethods of the continuousmaximumprinciple has not beenproved efficient [32].
Accordingly, theapplicationofOPCtoscheduling is important for tworeasons. First, aconceptual
problemconsistsof thecontinuousvaluesof thecontrolvariables. Second,acomputationalproblem
with adirectmethodof themaximumprinciple exists. These shortcomings set limitations on the
applicationofOPCtopurelycombinatorialproblems.
Todate,variousdynamicmodels,methodsandalgorithmshavebeenproposedtosolveplanning
andschedulingproblems indifferentapplicationareas ([13,19,33–38]). In thesepapers, transformation
procedures from classical scheduling models to their dynamic analogue have been developed.
Forexample, the followingmodelsofOPCwereproposed:
Mg—dynamicmodelofMSelementsandsubsystemsmotioncontrol;
Mk—dynamicmodelofMSchannelcontrol;
Mo—dynamicmodelofMSoperationscontrol;
Mf—dynamicmodelofMSflowcontrol;
Mp—dynamicmodelofMSresourcecontrol;
Me—dynamicmodelofMSoperationparameterscontrol;
Mc—dynamicmodelofMSstructuredynamiccontrol;
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Buch Algorithms for Scheduling Problems"
Algorithms for Scheduling Problems
- Titel
- Algorithms for Scheduling Problems
- Autoren
- Frank Werner
- Larysa Burtseva
- Yuri Sotskov
- Herausgeber
- MDPI
- Ort
- Basel
- Datum
- 2018
- Sprache
- englisch
- Lizenz
- CC BY 4.0
- ISBN
- 978-3-03897-120-7
- Abmessungen
- 17.0 x 24.4 cm
- Seiten
- 212
- Schlagwörter
- Scheduling Problems in Logistics, Transport, Timetabling, Sports, Healthcare, Engineering, Energy Management
- Kategorien
- Informatik
- Technik