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Starting with the description of the Traveling Salesmen Problem formulation as given by van Vyve and Wolsey in the article Approximate extended formulations'', we investigate the effects of small variations onto the performance of contemporary mixed integer programming solvers. We will show that even minor changes in the formulation of the model can result in performance difference of more than a factor of 1000. As the results show it is not obvious which changes will result in performance improvements and which not.
In \emph{classical optimization} it is assumed that full information about the problem to be solved is given. This, in particular, includes that all data are at hand. The real world may not be so nice'' to optimizers. Some problem constraints may not be known, the data may be corrupted, or some data may not be available at the moments when decisions have to be made. The last issue is the subject of \emph{online optimization} which will be addressed here. We explain some theory that has been developed to cope with such situations and provide examples from practice where unavailable information is not the result of bad data handling but an inevitable phenomenon.
Ein gemischt-ganzzahliges lineares Optimierungsmodell für ein Laserschweißproblem im Karosseriebau
(2006)
Ein gemischt-ganzzahliges lineares Optimierungsmodell für ein Laserschweißproblem im Karosseriebau
(2006)
Wir beschäftigen uns mit dem Problem der Betriebsplanung von Laserschweißrobotern im Karosseriebau. Gegeben ist eine Menge von Schweißnähten, die innerhalb einer Fertigungszelle an einem Karosserieteil gefertigt werden müssen. Die Schweißnähte werden durch mehrere parallel betriebene Roboter bearbeitet. Die Aufgabe besteht darin, für jeden Roboter eine Reihenfolge und eine zeitliche Koordinierung seiner Bewegungen zu finden, so dass alle Schweißnähte innerhalb der Taktzeit der Fertigungszelle bearbeitet werden und so wenig Laserquellen wie möglich eingesetzt werden. Dabei müssen einige Nebenbedingungen berücksichtigt werden. Für dieses spezielle Schweißproblem haben wir eine Formulierung als gemischt-ganzzahliges lineares Programm entwickelt, welches sich für die untersuchten praktischen Fälle sehr schnell lösen lässt.
The standard computational methods for computing the optimal value functions of Markov Decision Problems (MDP) require the exploration of the entire state space. This is practically infeasible for applications with huge numbers of states as they arise, e.\,g., from modeling the decisions in online optimization problems by MDPs. Exploiting column generation techniques, we propose and apply an LP-based method to determine an $\varepsilon$-approximation of the optimal value function at a given state by inspecting only states in a small neighborhood. In the context of online optimization problems, we use these methods in order to evaluate the quality of concrete policies with respect to given initial states. Moreover, the tools can also be used to obtain evidence of the impact of single decisions. This way, they can be utilized in the design of policies.
Many online problems encountered in real-life involve a two-stage decision process: upon arrival of a new request, an irrevocable first-stage decision (the assignment of a specific resource to the request) must be made immediately, while in a second stage process, certain ``subinstances'' (that is, the instances of all requests assigned to a particular resource) can be solved to optimality (offline) later. We introduce the novel concept of an \emph{Online Target Date Assignment Problem} (\textsc{OnlineTDAP}) as a general framework for online problems with this nature. Requests for the \textsc{OnlineTDAP} become known at certain dates. An online algorithm has to assign a target date to each request, specifying on which date the request should be processed (e.\,g., an appointment with a customer for a washing machine repair). The cost at a target date is given by the \emph{downstream cost}, the optimal cost of processing all requests at that date w.\,r.\,t.\ some fixed downstream offline optimization problem (e.\,g., the cost of an optimal dispatch for service technicians). We provide general competitive algorithms for the \textsc{OnlineTDAP} independently of the particular downstream problem, when the overall objective is to minimize either the sum or the maximum of all downstream costs. As the first basic examples, we analyze the competitive ratios of our algorithms for the par ticular academic downstream problems of bin-packing, nonpreemptive scheduling on identical parallel machines, and routing a traveling salesman.
The task of an elevator control is to schedule the elevators of a group such
that small waiting and travel times for the passengers are obtained. We present an exact
reoptimization algorithm for this problem. A reoptimization algorithm computes a
new schedule for the elevator group each time a new passenger arrives. Our algorithm
uses column generation techniques and is, to the best of our knowledge, the first exact
reoptimization algorithms for a group of passenger elevators. To solve the column
generation problem, we propose a Branch & Bound method.