Online Optimization: Probabilistic Analysis and Algorithm Engineering
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Online Optimization: Probabilistic Analysis and Algorithm Engineering vorgelegt von Dipl.-Inf. Benjamin Hiller Von der Fakultät II – Mathematik und Naturwissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften – Dr. rer. nat. – genehmigte Dissertation Berichter: Prof. Dr. Dr. h.c. mult. Martin Grötschel Dr. Tjark Vredeveld Vorsitzender: Prof. Dr. Fredi Tröltzsch Tag der wissenschaftlichen Aussprache: 15. Dezember 2009 Berlin 2010 D 83 Zusammenfassung Diese Arbeit beschäftigt sich mit Online-Optimierung, also der Steuerung von Systemen, bei denen die für die optimierte Steuerung relevanten Daten erst mit der Zeit, d. h. online, bekannt werden. Wir konzentrieren uns dabei auf kombinatorische Online-Optimierungsprobleme, bei denen die Steuerungsentscheidungen diskret sind. Im ersten, praktisch orientierten Teil der Arbeit werden Reoptimie- rungsalgorithmen für die Online-Steuerung komplexer realer Systeme vorgestellt. Ein Reoptimierungsalgorithmus trifft seine Steuerungsent- scheidung so, dass sie in einem bestimmten Sinn “günstig” für die aktuelle Situation ist. Wir benutzen Techniken der mathematischen Optimierung, insbesondere ganzzahlige Optimierung, um fortgeschrit- tene Reoptimierungsalgorithmen zu entwickeln. Unsere erste Anwen- dung ist die automatische Disposition von Pannenhilfefahrzeugen des ADAC. Hier zeigt sich, dass mathematische Optimierungsmethoden den Dispositionsprozess gegenüber der Planung mit einfachen Heuristiken verbessern und kürzere Wartezeiten für die Kunden erzielen. Für die zweite Anwendung, die Steuerung von Gruppen von Personenaufzügen in Hochhäusern, entwickeln wir ebenfalls Reoptimierungsalgorithmen. Diese sind speziell auf die Steuerung von Zielrufsystemen, bei denen Passagiere ihre Zieletage bereits auf der Startetage eingeben, zugeschnit- ten. Zusätzlich zur Steuerung der Aufzugsgruppe unter Ausnutzung der Freiheitsgrade des Systems versuchen die Algorithmen, langfristig ungünstige Steuerungen zu vermeiden, was durch eine geeignete Model- lierung erreicht wird. Unsere Simulationen zeigen, dass die Anzahl der Passagiere, die mit akzeptabler Servicequalität bedient werden kann, durch die Verwendung von Zielrufsystemen um 50% gesteigert werden kann. Im theoretischen zweiten Teil stellen wir einen neuen Ansatz zur probabilistischen Analyse von Onlinealgorithmen vor. Im Gegensatz zu bestehenden Analysen bewerten wir die Güte eines Algorithmus nicht anhand des Erwartungswertes der Zielfunktion, sondern anhand der Verteilung der Zielfunktionswerte auf allen möglichen Eingaben, die eine globalere Beschreibung der Güte des Algorithmus darstellt. Mithilfe des Konzepts der stochastischen Dominanz können wir z. B. zeigen, dass der bekannte Paging-Algorithmus LRU eine bezüglich der stochastischen Dominanz optimale Verteilung erreicht, wenn die Eingabe bestimmte praktisch häufig vorkommende Lokalitätseigenschaften aufweist. Für das Online-Bin-Coloring-Problem [KdPSR01] beweisen wir eine Aussage, die das in Simulationen beobachtete Verhalten erklärt und damit eine offene Frage aus [KdPSR01] beantwortet. Abstract The subject of this thesis is online optimization, which deals with making decisions in an environment where the data describing the process to optimize becomes available over time, i. e., online. In particular, we study combinatorial online optimization problems involving discrete decisions both from a practical and a theoretical point of view. The first part of the thesis is devoted to reoptimization algorithms for the online control of complex real-world systems. A reoptimization algorithm obtains its online control decision by determining a decision that is in some sense “good” for the current state of the system. We apply rigorous mathematical modelling and optimization methods based on Integer Progamming to develop advanced reoptimization algorithms. Our first application concerns the automatic dispatching of a large fleet of service vehicles to serve waiting customers. We find that rigorous methods improve the performance of the dispatching process, leading to shorter waiting times for the customers. In our second application we consider the scheduling of groups of passenger elevators in high rise buildings. We suggest advanced control algorithms for destination call systems, in which a passenger enters his desired destination floor already at his current floor. In addition to exploiting all degrees of freedom offered by the system, our reoptimization algorithms feature means to avoid decisions that will lead to undesirable online behavior. Our simulation experiments indicate that the number of passengers that can be served with an acceptable service level increases by 50% by using a destination call system controlled by our algorithms instead of a conventional system. The second part introduces a novel kind of probabilistic analysis for online algorithms. In contrast to existing probabilistic analyses, we do not judge the quality of an online algorithm using the expectation of the objective. Instead, we consider the distribution of the objective value on all inputs which gives a more global description of the performance of the algorithm. Using the notion of stochastic dominance, we are able to establish that certain online algorithms obtain better objective value distributions than others. For instance, we can show that the famous paging algorithm LRU achieves a distribution that is optimal w. r. t. the stochastic dominance order if the request sequences exhibit locality of reference. We also apply this approach to the analysis of algorithms for online bin coloring [KdPSR01], obtaining a result that explains the behavior observed in simulations, thus resolving an open problem posed in [KdPSR01]. Acknowledgements The theoretical part of this thesis grew out of the project “Combinatorial online planning” which was part of the DFG research group “Algorithms, Structures, Randomness”. I want to thank my colleagues from this research group, in particular Arie Koster and Volker Kaibel, for inspiring discussions and encouraging my own research. Special thanks go to Tjark Vredeveld, with whom I started the analysis of the bin coloring algorithms and who always motivated me to continue the work. I am very grateful to him for hosting me for several stays in the nice city of Maastricht. Before even starting in the “Combinatorial online planning” project I did already work on the project with the ADAC together with Jörg Rambau and Sven O. Krumke. Both guided my research there, but also let me pursue own ideas. The ADAC project was a good preparation for two research projects with Kollmorgen Steuerungstechnik on the development of control al- gorithms for passenger elevators. I want to thank Björn Kollmorgen and Peter Gerstenmeyer, who provided important insights in the details of elevator scheduling. Most importantly, they believed that we young researchers could deliver insights and control algorithms that are at least as good as if Kollmorgen Steuerungstechnik developed them on their own. I have to mention Martin Grötschel’s way of supporting young researchers here: He just lets them do own projects, so one quickly learns to take responsibility, but also to define own research goals. Andreas Tuchscherer, who also worked on the elevator projects, was a big help both in very valuable discussions and in sharing the overtime work to meet the project deadlines. Finally, I have to thank my proof readers Torsten Klug, Jacint Szabó, and Andreas Tuchscherer for providing me with valuable feedback, despite the tight deadline I imposed on them. Their feedback, and in particular Andreas’ detailed comments and questions, helped a lot to improve the presentation and to wipe out errors. I also want to thank my other colleagues at ZIB for the nice working atmosphere and their encouragement during the completion of my thesis. Last not least I want to thank my wife Petra for her patience, in particular during the last two months when I did almost nothing else than to finish this work. Contents Introduction and overview 1 I. Design of reoptimization algorithms 5 1. Dispatching service vehicles for the ADAC under high load 7 1.1. Issues and motivation . .7 1.2. Simplified models . .9 1.2.1. The original ZIBDIP model . .9 1.2.2. The simplified model 4-ZIBDIP .................... 10 1.2.3. The simplified model PTC (Prescribed Total Cover) . 10 1.2.4. The simplified model ShadowPrice ................... 11 1.2.5. The simplified model ZIBDIPdummy .................. 12 1.3. Simplified reoptimization algorithms . 12 1.3.1. The simplified algorithm BestInsert .................. 12 1.3.2. The simplified algorithm 2-Opt .................... 12 1.4. Computational results . 13 1.4.1. Simplified models . 13 1.4.2. Simplified reoptimization algorithms . 15 1.5. Significance . 16 2. Group control of elevator systems 23 2.1. Conventional elevator systems vs. destination call systems . 24 2.2. Some more background on elevator control . 28 2.3. Assumptions and requirements for elevator dispatches . 31 2.3.1. General requirements . 31 2.3.2. Additional assumptions . 33 2.3.3. System-specific requirements . 34 2.4. A general model for elevator group control . 35 2.4.1. The structure of the snapshot problem . 36 2.4.2. A model for elevator schedules . 37 2.5. Heuristic algorithms for group elevator control . 42 2.5.1. Classical elevator control . 42 2.5.2. Computer Group Control . 44 2.5.3. Genetic algorithms . 44 2.5.4. A cost-based best-insertion heuristic . 44 i Contents 3. Exact elevator group