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Scheduling under Uncertainty : Solution Approaches

Scheduling under Uncertainty : Solution Approaches . Frank Werner Faculty of Mathematics. Outline of the talk. Introduction Stochastic approach Fuzzy approach Robust approach Stability approach Selection of a suitable approach. 1. Introduction. Notations.

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Scheduling under Uncertainty : Solution Approaches

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  1. SchedulingunderUncertainty:Solution Approaches Frank Werner FacultyofMathematics

  2. Outline ofthe talk • Introduction • Stochasticapproach • Fuzzyapproach • Robust approach • Stabilityapproach • Selectionof a suitableapproach

  3. 1. Introduction Notations

  4. Deterministicmodels:all dataaredeterministicallygiven in advance • Stochasticmodels:dataincluderandom variables In real-lifescheduling: manytypesofuncertainty(e.g. processingtimes not exactlyknown, machinebreakdowns, additionallyarivingjobswithhighpriorities, roundingerrors, etc.) Uncertain (interval) processingtimes:

  5. Relationshipbetweenstochasticanduncertainproblems: Distribution function Densityfunction

  6. Approaches forproblemswithinaccuratedata: • Stochasticapproach: useofrandom variables withknownprobabilitydistributions • Fuzzyapproach: fuzzynumbersasdata • Robust approach: determinationof a schedulehedgingagainsttheworst-casescenario • Stabilityapproach: combinationof a stabilityanalysis, a multi-stagedecisionframeworkandtheconceptof a minimal dominant setof semi-activeschedules → Thereisnouniquemethodfor all typesofuncertainties.

  7. Two-phasedecision-makingprocedure • Off-line (proactive) phaseconstructionof a setofpotentially optimal solutionsbeforetherealizationoftheactivities(staticschedulingenvironment, scheduleplanningphase) • On-line (reactive) phaseselectionof a solutionfromwhenmoreinformationisavailableand/or a partoftheschedulehasalreadybeenrealized→ useof fast algorithms(dynamicschedulingenvironment, scheduleexecutionphase)

  8. General literature (surveys) • Pinedo: Scheduling, Theory, Algorithmsand Systems, Prentice Hall, 1995, 2002, 2008, 2012 • SlowinskiandHapke: SchedulingunderFuzziness, Physica, 1999 • Kasperski: DiscreteOptimizationwithInterval Data, Springer, 2008 • Sotskov, Sotskova, Lai and Werner: SchedulingunderUncertainty; TheoryandAlgorithms, Belarusian Science, 2010 Forthe RCPSP underuncertainty, see e.g. • HerroelenandLeus, Int. J. Prod. Res.. 2004 • HerroelenandLeus, EJOR, 2005 • DemeulemeesterandHerroelen, Special Issue, J. Scheduling, 2007

  9. 2. Stochasticapproach • Distribution ofrandom variables(e.g. processingtimes, releasedates, due dates)known in advance • Often: minimizationofexpectationvalues(ofmakespan, total completion time, etc.) Classesofpolicies(seePinedo 1995) • Non-preemptivestaticlistpolicy (NSL) • Preemptivestaticlistpolicy (PSL) • Non-preemptivedynamicpolicy (ND) • Preemptivedynamicpolicy (PD)

  10. Someresultsforsingle-stageproblems(seePinedo 1995) Single machineproblems • ProblemWSEPT rule: order thejobsaccordingto non-increasingratiosTheorem 1: The WSEPT ruledetermines an optimal solution in theclassof NSL as well as ND policies. • Problem Theorem 2: The EDD ruledetermines an optimal solution in theclassof NSL, ND and PD policies.

  11. Problem Theorem 3: The WSEPT ruledetermines an optimal solution in theclassof NSL, ND and PD policies.Remark: The same resultholdsforgeometricallydistributed Parallel machineproblems • ProblemTheorem 4: The LEPT ruledetermines an optimal solution in theclassof NSL policies.

  12. ProblemTheorem 5: The non-preemptive LEPT policydetermines an optimal solution in theclassof PD policies. • ProblemTheorem 6: The non-preemptive SEPT policydetermines an optimal solution in theclassof PD policies.

  13. Selected references (1) • PinedoandWeiss, Nav. Res. Log. Quart., 1979 • Glazebrook, J. Appl. Prob., 1979 • WeissandPinedo, J. Appl. Prob., 1980 • Weber, J. Appl. Prob., 1982 • Pinedo, Oper. Res., 1982; 1983 • Pinedo, EJOR, 1984 • PinedoandWeiss, Oper. Res., 1984 • Möhring, Radermacher andWeiss, ZOR, 1984; 1985 • Pinedo, Management Sci., 1985 • Wie andPinedo, Math. Oper. Res., 1986 • Weber, VaraiyaandWalrand, J. Appl. Prob., 1986 • Righter, System andControl Letters, 1988 • Weiss, Ann. Oper. Res., 1990

  14. Selected references (2) • Weiss, Math. Oper. Res., 1992 • Righter, Stochastic Orders, 1994 • Cai and Tu, Nav. Res. Log., 1996 • Cai and Zhou, Oper. Res., 1999 • Möhring, Schulz andUetz, J. ACM, 1999 • Nino-Mora, Encyclop. Optimiz., 2001 • Cai, Sun and Zhou, Prob. Eng. Inform. Sci., 2003 • Ebben, Hans andOldeWeghuis, OR Spectrum, 2005 • Ivanescu, Fransooand Bertrand, OR Spectrum, 2005 • Cai, Wu and Zhou, IEEE Transactions Autom. Sci. Eng., 2007 • Cai, Wu and Zhou, J. Scheduling, 2007; 2011 • Cai, Wu and Zhou, Oper. Res., 2009 • Tam, Ehrgott, Ryan andZakeri, OR Spectrum, 2011

  15. 3. Fuzzyapproach • Fuzzyschedulingtechniqueseitherfuzzifyexistingschedulingrulesorsolvemathematicalprogrammingproblems • Often: fuzzyprocessingtimes , fuzzy due dates • Examplestriangularfuzzyprocessingtimestrapezoidalfuzzyprocessingtimes

  16. Often: possibilisticapproach(Dubois andPrade 1988) Chanasand Kasperski (2001) Problem Objective: Assumption: → adaptionofLawler‘salgorithmforproblem

  17. Special cases: Alternative goalapproach - fuzzygoal, Objective: Chanasand Kasperski (2003) Problem Objective: → adaptionofLawler‘salgorithmforproblem

  18. Selected references (1) • Dumitru andLuban, Fuzzy Sets and Systems, 1982 • Tada, Ishii andNishida, APORS, 1990 • Ishii, TadaandMasuda, Fuzzy Sets and Systems, 1992 • GrabotandGeneste, Int. J. Prod. Res., 1994 • Han, Ishii and Fuji, EJOR, 1994 • Ishii andTada, EJOR, 1995 • Stanfield, King andJoines, EJOR, 1996 • Kuroda and Wang, Int. J. Prod. Econ., 1996 • Özelkanand Duckstein, EJOR, 1999 • SakawaandKubota, EJOR, 2000

  19. Selected references (2) • Chanasand Kasperski, Eng. Appl. Artif. Intell., 2001 • Chanasand Kasperski, EJOR, 2003 • Chanasand Kasperski, Fuzzy Sets and Systems, 2004 • Itohand Ishii, FuzzyOptim. andDec. Mak., 2005 • Kasperski, Fuzzy Sets and Systems, 2005 • Inuiguchi, LNCS, 2007 • Petrovic, Fayad, Petrovic, Burke and Kendall, Ann. Oper. Res., 2008

  20. 4. Robust approach Objective: Find a solution, whichminimizesthe „worst-case“ performanceover all scenarios. Notations (singlemachineproblems) maximal regretof Minmaxregretproblem (MRP): Find a sequence such that

  21. Somepolynomiallysolvable MRP (Kasperski 2005) (VolgenantandDuin 2010) (Averbakh 2006) (Kasperski 2008) Some NP-hard MRP (LebedevandAverbakh 2006) (for a 2-approximation algorithm, see Kasperski and Zielinski 2008) (Kasperski, Kurpiszand Zielinski 2012)

  22. Kasperski andZielinski (2011) ConsiderationofMRP‘susingfuzzyintervals Deviation interval Known: deviation Applicationofpossibilitytheory(Dubois andPrade 1988) possibly optimal if necessarily optimal if

  23. Fuzzyproblem orequivalently whereis a fuzzyintervalandisthecomplementofwithmembershipfunction The fuzzyproblemcanbeefficientlysolvedif a polynomialalgorithmforthecorresponding MRP exists.

  24. Solution approaches • Binary searchmethod- repeatedexactsolutionofthe MRP - applications: : binarysearchsubroutine in B&B algorithm

  25. Mixed integer programmingformulation- useof a MIP solver- application: • Parametricapproach- solutionof a parametricversionof a MRP(often time-consuming)- application:

  26. Selected references (1) • Daniels andKouvelis, Management Sci., 1995 • KouvelisandYu, Kluwer, 1997 • Kouvelis, Daniels andVairaktarakis, IEEE Transactions, 2000 • Averbakh, OR Letters, 2001 • Yang andYu, J. Comb. Optimiz., 2002 • Kasperski, OR Letters, 2005 • Kasperski and Zielinski, Inf. Proc. Letters, 2006 • LebedevandAverbakh, DAM, 2006 • Averbakh, EJOR, 2006 • Montemanni, JMMA, 2007

  27. Selected references (2) • Kasperski and Zielinski, OR Letters, 2008 • Sabuncuogluand Goren, Int. J. Comp. Integr. Manufact., 2009 • Aissi, BazganandVanderpooten, EJOR, 2009 • VolgenantandDuin, COR, 2010 • Kasperski and Zielinski, FUZZ-IEEE, 2011 • Kasperski, Kurpiszand Zielinski, EJOR, 2012

  28. 5. Stabilityapproach 5.1. Foundations 5.2. General shopproblem 5.3. Two-machineflowandjobshopproblems 5.4. Problem

  29. 5.1. Foundations Mixed Graph Example: 11 12 13 00 ** 21 22 23

  30. Example (continued) 11 12 13 ** 00 21 22 23

  31. Stabilityanalysisof an optimal digraph Definition 1 The closed ball iscalled a stability ball ofifforanyremains optimal. The maximal value iscalledthestabilityradiusofdigraph Known: • Characterizationofthe extreme valuesof • Formulasforcalculating • Computationalresultsforjobshopproblemswith(seeSotskov, Sotskovaand Werner, Omega, 1997)

  32. 5.2. General shopproblem Definition 2 iscalled a G-solutionforproblemifforanyfixedcontains an optimal digraph. Ifanyis not a G-solution, iscalled a minimal G-solutiondenotedas Introductionofthe relative stabilityradius:

  33. Definition 3 Letbe such thatforanyThe maximal valueofof such a stability ball iscalledtherelative stabilityradius Known: • Dominancerelationsamongpathsandsetsofpaths • Characterizationofthe extreme valuesof

  34. Characterizationof a G-solutionforproblem Definition 4 (strongly) dominatesin → dominancerelation Theorem 7: is a G-solution. Thereexists a finite coveringofpolytopebyclosedconvexsetswith such thatforanyandanythereexists a forwhich Corollary:

  35. Theorem 8: Letbe a G-solutionwith Then: is a minimal G-solution. Foranythereexists a vector such that Algorithmsforproblem

  36. Several 3-phase schemes: • B&B: implicit (or explicit) enumerationschemeforgenerating a G-solution • SOL: reductionofbygenerating a sequencewiththe same and different • MINSOL: generationof a minimal G-solutionby a repeatedapplicationofalgorithm SOL

  37. Somecomputationalresults: Exact sol.: , Heuristic sol.:

  38. 5.3. Two-machineproblemswithintervalprocessingtimes • ProblemJohnson permutation:Partition ofthejobset

  39. Theorem 9: • foranyeither (either ) and • andifsatisfies

  40. Theorem 10: If then Percentageofinstanceswith , where

  41. General caseofproblem Theorem 11: Thereexists an Theorem 12:

  42. Example: without transitive arcs: J2 J1 J3 J5 J6 J4

  43. Propertiesof in thecaseofseeMatsveichuk, Sotskovand Werner, Optimization, 2011 Schedule executionphase:seeSotskov, Sotskova, Lai and Werner, Schedulingunderuncertainty, 2010 (Section 3.5) Computationalresultsforandfor • Problem→ Reductiontotwoproblems:seeSotskov, Sotskova, Lai and Werner, Schedulingunderuncertainty, 2010 (Section 3.6)

  44. 5.4. Problem Notations:

  45. Definition ofthestability box:

  46. Definition 5 The maximal closedrectangular box is a stability box ofpermutation , ifpermu-tationbeing optimal forinstancewith a scenarioremains optimal fortheinstancewith a scenarioforeachIftheredoes not exist a scenario such thatpermutationis optimal forinstance , then Remark: The stability box is a subsetofthestabilityregion. However, thestability box isusedsinceitcaneasilybecomputed.

  47. Theorem 13: Fortheproblem , jobdominatesifandonlyifthefollowinginequalityholds: Lower (upper) boundon therangeofpreservingtheoptimalityof :

  48. Theorem 14: Ifthereisnojob , in permutation such thatinequality holdsforat least onejob , thenthestability box iscalculatedasfollows: otherwise

  49. Example: Data forcalculating

  50. Stability box for Relative volumeof a stability box Maximal rangesofpossiblevariationsoftheprocessingtimes , withinthestability box aredashed.

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