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In this thesis we describe a practical problem that we encountered in the on--line optimization of a complex Flexible Manufacturing System. In the considered system a stacker crane has to fulfill all transportation tasks (jobs) in a single aisled automatic storage system. The jobs have to be sequenced in such a way, that the time needed for the unloaded moves is minimized. The modelling of this question leads to the so--called on--line Hamiltonian path problem. We computationally compare several on--line heuristics and derive lower bounds on the value obtained by an optimal on--line strategy by analyzing two off--line Combinatorial Optimization problems: the asymmetric Hamiltonian path problem with precedence constraints, also called sequential ordering problem (SOP), and the asymmetric Hamiltonian path problem with time windows (AHPPTW). We study the SOP and AHPPTW from a polyhedral point of view and derive several new classes of valid inequalities. Based on the polyhedral investigations we develop branch&cut algorithms for both problems and can achieve encouraging results on solving problem instances from real--world examples of the practical application.
Many optimization problems have several equivalent mathematical models. It is often not apparent which of these models is most suitable for practical computation, in particular, when a certain application with a specific range of instance sizes is in focus. Our paper addresses the Asymmetric Travelling Salesman Problem with time windows (ATSP-TW) from such a point of view. The real--world application we aim at is the control of a stacker crane in a warehouse. We have implemented codes based on three alternative integer programming formulations of the ATSP-TW and more than ten heuristics. Computational results for real-world instances with up to 233 nodes are reported, showing that a new model presented in a companion paper outperforms the other two models we considered --- at least for our special application --- and that the heuristics provide acceptable solutions.
In ``classical'' optimization, all data of a problem instance are considered given. The standard theory and the usual algorithmic techniques apply to such cases only. Online optimization is different. Many decisions have to be made before all data are available. In addition, decisions once made cannot be changed. How should one act ``best'' in such an environment? In this paper we survey online problems coming up in combinatorial optimization. We first outline theoretical concepts, such as competitiveness against various adversaries, to analyze online problems and algorithms. The focus, however, lies on real-world applications. We report, in particular, on theoretical investigations and our practical experience with problems arising in transportation and the automatic handling of material.
The asymmetric travelling salesman problem with time windows (ATSP-TW) is a basic model for scheduling and routing applications. In this paper we present a formulation of the problem involving only 0/1-variables associated with the arcs of the underlying digraph. This has the advantage of avoiding additional variables as well as the associated (typically very ineffective) linking constraints. In the formulation, time window restrictions are modelled by means of ``infeasible path elimination'' constraints. We present the basic form of these constraints along with some possible strengthenings. Several other classes of valid inequalities derived from related asymmetric travelling salesman problems are also described, along with a lifting theorem. We also study the ATSP-TW polytope, $P_{TW}$, defined as the convex hull of the integer solutions of our model. We show that determining the dimension of $P_{TW}$ is strongly {\em NP}--complete problem, even if only one time window is present. In this latter case, we provide a minimal equation system for $P_{TW}$. Computational experiments on the new formulation are reported in a companion paper [1997] where we show that it outperforms alternative formulations on some classes of problem instances.
Die Business Unit PC in Augsburg ist die zentrale Produktionsstätte der Siemens--Nixdorf Informationssysteme (SNI) AG für Personal Computer sowie für einige Periphärgeräte. Das Werk, entworfen nach modernen CIM/CAI--Konzepten (Computer Integrated Manufacturing/ Computer Aided Industry), wurde 1987 errichtet. Bald zeigte sich jedoch, daß es für ein zu geringes Produktionsvolumen ausgelegt war und einige Komponenten des Systems Engpässe im Produktionsbetrieb darstellen. Das Management suchte nach Möglichkeiten, den Produktionsfluß zu verbessern, ohne teure technische Änderungen am System vornehmen zu müssen. Eine Forschungsgruppe des Konrad--Zuse--Zentrums für Informationstechnik (die ehemals an der Universität Augsburg ansässig war) analysierte, unterstützt von einigen Studenten und Ingenieuren der SNI, den Produktionsfluß und lokalisierte Schwachstellen. Basierend auf diesen Erkenntnissen wurden mathematische Fragestellungen erarbeitet und auf mathematischen Optimierungsverfahren basierende Softwarepakete entwickelt, die jetzt teilweise bei SNI im Einsatz sind. Im folgenden werden einige dieser Fragestellungen, deren Modellierung und mathematische Behandlung beschrieben. Einige der Ansätze, die hier dargestellt werden sollen, sind teilweise schon in Grötschel [Grö92] angesprochen worden.
We report on a joint project with industry that had the aim to sequence transportation requests within an automatic storage system in such a way that the overall travel time is minimized. The manufacturing environment is such that scheduling decisions have to be made before all jobs are known. We have modeled this task as an \emph{online} Asymmetric Traveling Salesman Problem (ATSP). Several heuristics for the online ATSP are compared computationally within a simulation environment to judge which should be used in practice. Compared to the priority rule used so far, the optimization package reduced the unloaded travel time by about 40~\%. Because of these significant savings our procedure was implemented as part of the control software for the stacker cranes of the storage systems.
A Branch & Cut Algorithm for the Asymmetric Traveling Salesman Problem with Precedence Constraints
(1997)
In this paper we consider a variant of the classical ATSP, namely the asymmetric Hamiltonian path problem (or equivalently ATSP) with precedence constraints. In this problem precedences among the nodes are present, stating that a certain node has to precede others in any feasible sequence. This problem occurs as a basic model in scheduling and routing and has a wide range of applications varying from helicopter routing[Timlin89], sequencing in flexible manufacturing [AscheuerEscuderoGroetschelStoer90,AscheuerEscuderoGroetschelStoer93], to stacker crane routing in an automatic storage system[Ascheuer95]. We give an integer programming model and summarize known classes of valid inequalities. We describe in detail the implementation of a branch&-cut algorithm and give computational results on real world instances and benchmark problems from TSPLIB. The results we achieve indicate that our implementation outperforms other implementations found in the literature. Real world instances up to 174 nodes could be solved to optimality within a few minutes of CPU-time. As a side product we obtained a branch&cut-algorithm for the ATSP. All instances in TSPLIB could be solved to optimality in a reasonable amount of computing time.
This paper gives a short introduction into combinatorial online optimization. It explains a few evaluation concepts of online algorithms, such as competitiveness, and discusses limitations in their application to real--world problems. The main focus, however, is a survey of combinatorial online problems coming up in practice, in particular, in large scale material flow and flexible manufacturing systems.