We investigate a novel scheduling problem which is motivated by an
application in the Australian railway industry. Given a set of maintenance jobs
and a set of train paths over a railway corridor with bidirectional traffic, we
seek a schedule of jobs such that a minimum number of train paths are cancelled
due to conflict with the job schedule. We show that the problem is NP-complete
in general. In a special case of the problem when every job under any schedule
just affects one train path, and the speed of trains is bounded from above and
below, we show that the problem can be solved in polynomial time. Moreover, in
another special case of the problem where the traffic is unidirectional, we
show that the problem can be solved in time O(n4)
This thesis is a collection of papers on a variety of optimization problems where network structure can be used to obtain efficient algorithms. The considered applications range from the optimization of radiation treatment plkans in cancer therapy to maintenance planning for maximizing the throughput in bulk good supply chains
This thesis explores possibilities and limitations of extending classical combinatorial optimization problems for network flows and network design. We propose new mathematical models for logistics networks that feature commodities with multidimensional properties, e.g. their mass and volume, to capture consolidation effects of commodities with complementing properties. We provide new theoretical insights and solution methods with immediate practical impact that we test on real-world instances from the automotive, chemical, and retail industry.
The first model is for tactical transportation planning with temporal consolidation effects. We propose various heuristics and prove for our instances, that most of our solutions are within a single-digit percentage of the optimum. We also study problem variants where commodities are routed unsplittably and give hardness results for various special cases and a dynamic program that finds optimal forest solutions, which overestimate real costs.
The second model is for strategic route planning under uncertainty. We provide for a robust optimization method that anticipates fluctuations of demands by minimizing worst-case costs over a restricted scenario set. We show that the adversary problem is NP-hard. To still find solutions with very good worst-case cost, we derive a carefully relaxed and simplified MILP, which solves well for large instances. It can be extended to include hub decisions leading to a robust M-median hub location problem. We find a price of robustness for our instances that is moderate for scenarios using average demand values as lower bounds. Trend based scenarios show a considerable tradeoff between historical average costs and worst case costs.
Another robustness concept are incremental hub chains that provide solutions for every number of hubs to operate, such that they are robust under changes of this number. A comparison of incremental solutions with M-median solutions obtained with an LP-based search suggests that a price of being incremental is low for our instances.
Finally, we investigate the problem of scheduling the maintenance of edges in a network. We focus on maintaining connectivity between two nodes over time. We show that the problem can be solved in polynomial time in arbitrary networks if preemption is allowed. If preemption is restricted to integral time points, the problem is NP-hard and for the non-preemptive case, we show strong non-approximability results.Diese Arbeit untersucht Möglichkeiten, klassische kombinatorische Optimierungsprobleme fĂŒr NetzwerkflĂŒsse und Netzwerkdesign zu erweitern. Wir stellen neue mathematische Modelle fĂŒr Logistiknetzwerke vor, die mehrdimensionale Eigenschaften der GĂŒter berĂŒcksichtigen, etwa Masse oder Volumen, um Konsolidierungseffekte von GĂŒtern mit komplementĂ€ren Eigenschaften zu nutzen. Wir erarbeiten neue theoretische Einsichten und Lösungsmethoden von praktischer Relevanz, die wir an realen Instanzen aus der Automobilindustrie, der Chemiebranche und aus dem Einzelhandel evaluieren. FĂŒr die taktische Transportplanung mit zeitlichen Konsolidierungseffekte erarbeiten wir verschiedene Heuristiken, welche fĂŒr unsere Instanzen die OptimalitĂ€tslĂŒcke zu 10% schlieĂen. Wir geben HĂ€rteresultate fĂŒr verschiedene SpezialfĂ€lle mit unteilbaren GĂŒtern an, sowie ein dynamisches Programm, welches Lösungen mit optimalen Baumkosten berechnet; eine ĂberschĂ€tzung der realen Kosten. FĂŒr die strategische Routenplanung unter Unsicherheit entwickeln wir eine robuste Optimierungsmethode, welche Nachfrageschwankungen antizipiert, indem Worstcase-Kosten ĂŒber einer beschrĂ€nkten Szenarienmenge minimiert werden. Wir zeigen, dass das Gegenspielerproblem NP-schwer ist. Um Lösungen mit guten Worstcase-Kosten zu finden, leiten wir ein sorgfĂ€ltig relaxiertes MILP her. Seine natĂŒrliche Erweiterung fĂŒr Hubentscheidungen fĂŒhrt auf ein robustes M-Median Hub Location Problem. Wir finden einen moderaten Preis der Robustheit fĂŒr Szenarien, die Durchschnittsnachfragemengen als untere Intervallgrenze verwenden. Trendbasierten Szenarien zeigen einen deutlichen Tradeoff zwischen historischen Durchschnittskosten und Worstcase-Kosten. Ein weiteres Robustheitskonzept stellen inkrementale Hubketten dar, welche Lösungen fĂŒr jede Anzahl an Hubstandorten angeben, sodass sie gegen Ănderungen dieser Anzahl robust sind. Ein Vergleich mit entsprechenden M-Median Lösungen, die wir mit einer LP-basierten Hubsuche erhalten, zeigt einen geringen Preis der InkrementalitĂ€t bei unseren Instanzen auf. Zuletzt untersuchen wir das Problem Wartungsarbeiten an Kanten in einem Netzwerk zu planen, um KonnektivitĂ€t zwischen zwei Knoten zu bewahren. Wir zeigen, dass sich das Problem polynomiell in beliebigen Netzen lösen lĂ€sst, falls Wartungsarbeiten unterbrochen werden dĂŒrfen. Falls dies nur zu ganzzahligen Zeitpunkten erlaubt ist, ist es bereits NP-schwer. FĂŒr den Fall ohne Unterbrechungen zeigen wir starke Nichtapproximierbarkeitsresultate
The symposium presented issues involved in the development of scheduling systems that can deal with resource and time limitations. To qualify, a system must be implemented and tested to some degree on non-trivial problems (ideally, on real-world problems). However, a system need not be fully deployed to qualify. Systems that schedule actions in terms of metric time constraints typically represent and reason about an external numeric clock or calendar and can be contrasted with those systems that represent time purely symbolically. The following topics are discussed: integrating planning and scheduling; integrating symbolic goals and numerical utilities; managing uncertainty; incremental rescheduling; managing limited computation time; anytime scheduling and planning algorithms, systems; dependency analysis and schedule reuse; management of schedule and plan execution; and incorporation of discrete event techniques
In this thesis we approach several problems with approximation algorithms; these are feasibility problems as well as optimization problems. In Chapter 1 we give a brief introduction into the general paradigm of approximation algorithms, motivate the problems, and give an outline of the
thesis. In Chapter 2, we discuss two algorithms to approximately generate a feasible solution of the mixed packing and covering problem which is
a model from convex optimization. This problem includes a large class of
linear programs. The algorithms generate approximately feasible solutions
within O(M(ln M+epsilon^{-2} ln epsilon^{-1})) and
O(M epsilon{-2} ln (M epsilon^{-1}))iterations,respectively,whereineachiterationablockproblemwhichdependsonthespecificapplicationhastobesolved.Bothalgorithms,appliedtolinearprograms,canresultincolumngenerationalgorithms.InChapter3,weimplementanalgorithmforthesoâcalledmaxâminâresourcesharingproblem.Thisisacertainconvexoptimizationproblemwhich,similartotheprobleminChapter1,includesalargeclassoflinearprograms.Theimplementation,whichisincludedintheappendix,isdoneinC++.WeusetheimplementationinthecontextofanAFPTASforStripPackinginordertoevaluatedynamicoptimizationofaparameterinthealgorithm,namelythesteplengthusedforinterpolation.Wecompareourchoicetothestaticsteplengthproposedintheanalysisofthealgorithmandconcludethatdynamicoptimizationofthesteplengthsignificantlyreducesthenumberofiterations.InChapter4,westudytwocloselyrelatedschedulingproblems,namelynonâpreemptiveschedulingwithfixedjobsandschedulingwithnonâavailabilityforsequentialjobsonmidenticalmachinesunderthemakespanobjective,wheremisconstant.Forthefirstproblem,whichdoesnotadmitanFPTASunlessP=NP,weobtainanewPTAS.Forthesecondproblem,weshowthatasuitablerestriction(namelythepermanentavailabilityofonemachine)isnecessarytoobtainaboundedapproximationratio.Forthisrestriction,whichdoesnotadmitanFPTASunlessP=NP,wepresentaPTAS;wealsodiscussthecomplexityofvariousspecialcases.Intotal,theresultsarebasicallybestpossible.InChapter5,wecontinuethestudiesfromChapter4wherenowthenumbermofmachinesispartoftheinput,whichmakestheproblemalgorithmicallyharder.Schedulingwithfixedjobsdoesnotadmitanapproximationratiobetterthan3/2,unlessP=NP;hereweobtainanapproximationratioof3/2+epsilonforanyepsilon>0.Forschedulingwithnonâavailability,werequireaconstantpercentageofthemachinestobepermanentlyavailable.Thisrestrictionalsodoesnotadmitanapproximationratiobetterthan3/2unlessP=NP;wealsoobtainanapproximationratioof3/2+\epsilon$ for any epsilon>0. With an interesting argument, the approximation ratio for both problems is
refined to exactly 3/2. We also point out an interesting relation of scheduling with fixed jobs to Bin Packing. As in Chapter 4, the results are in a certain sense best possible.
Finally, in Chapter 6, we conclude
with some remarks and open research problems
In this thesis we approach several problems with approximation algorithms; these are feasibility problems as well as optimization problems. In Chapter 1 we give a brief introduction into the general paradigm of approximation algorithms, motivate the problems, and give an outline of the
thesis. In Chapter 2, we discuss two algorithms to approximately generate a feasible solution of the mixed packing and covering problem which is
a model from convex optimization. This problem includes a large class of
linear programs. The algorithms generate approximately feasible solutions
within O(M(ln M+epsilon^{-2} ln epsilon^{-1})) and
O(M epsilon{-2} ln (M epsilon^{-1}))iterations,respectively,whereineachiterationablockproblemwhichdependsonthespecificapplicationhastobesolved.Bothalgorithms,appliedtolinearprograms,canresultincolumngenerationalgorithms.InChapter3,weimplementanalgorithmforthesoâcalledmaxâminâresourcesharingproblem.Thisisacertainconvexoptimizationproblemwhich,similartotheprobleminChapter1,includesalargeclassoflinearprograms.Theimplementation,whichisincludedintheappendix,isdoneinC++.WeusetheimplementationinthecontextofanAFPTASforStripPackinginordertoevaluatedynamicoptimizationofaparameterinthealgorithm,namelythesteplengthusedforinterpolation.Wecompareourchoicetothestaticsteplengthproposedintheanalysisofthealgorithmandconcludethatdynamicoptimizationofthesteplengthsignificantlyreducesthenumberofiterations.InChapter4,westudytwocloselyrelatedschedulingproblems,namelynonâpreemptiveschedulingwithfixedjobsandschedulingwithnonâavailabilityforsequentialjobsonmidenticalmachinesunderthemakespanobjective,wheremisconstant.Forthefirstproblem,whichdoesnotadmitanFPTASunlessP=NP,weobtainanewPTAS.Forthesecondproblem,weshowthatasuitablerestriction(namelythepermanentavailabilityofonemachine)isnecessarytoobtainaboundedapproximationratio.Forthisrestriction,whichdoesnotadmitanFPTASunlessP=NP,wepresentaPTAS;wealsodiscussthecomplexityofvariousspecialcases.Intotal,theresultsarebasicallybestpossible.InChapter5,wecontinuethestudiesfromChapter4wherenowthenumbermofmachinesispartoftheinput,whichmakestheproblemalgorithmicallyharder.Schedulingwithfixedjobsdoesnotadmitanapproximationratiobetterthan3/2,unlessP=NP;hereweobtainanapproximationratioof3/2+epsilonforanyepsilon>0.Forschedulingwithnonâavailability,werequireaconstantpercentageofthemachinestobepermanentlyavailable.Thisrestrictionalsodoesnotadmitanapproximationratiobetterthan3/2unlessP=NP;wealsoobtainanapproximationratioof3/2+\epsilon$ for any epsilon>0. With an interesting argument, the approximation ratio for both problems is
refined to exactly 3/2. We also point out an interesting relation of scheduling with fixed jobs to Bin Packing. As in Chapter 4, the results are in a certain sense best possible.
Finally, in Chapter 6, we conclude
with some remarks and open research problems
We consider a variant of the well-known single node fixed charge network flow set with constant capacities. This set arises from the relaxation of more general mixed integer sets such as lot-sizing problems with multiple suppliers. We provide a complete polyhedral characterization of the convex hull of the given set
We study the problem of scheduling maintenance on arcs of a capacitated network so as to maximize the total flow from a source node to a sink node over a set of time periods. Maintenance on an arc shuts down the arc for the duration of the period in which its maintenance is scheduled, making its capacity zero for that period. A set of arcs is designated to have maintenance during the planning period, which will require each to be shut down for exactly one time period. In general this problem is known to be NP-hard, and several special instance classes have been studied. Here we propose an additional constraint which limits the number of maintenance jobs per time period, and we study the impact of this on the complexity