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On the construction of optimal paths from flows and the analysis of evacuation scenarios

  • In Part I: "The flow-decomposition problem", we introduce and discuss the flow-decomposition problem. Given a flow F, this problem consists of decomposing the flow into a set of paths optimizing specific properties of those paths. We introduce different types of decompositions, such as integer decompositions and alpha-decompositions, and provide two formulations of the set of feasible decompositions. We show that the problem of minimizing the longest path in a decomposition is NP-hard, even for fractional solutions. Then we develop an algorithm based on column generation which is able to solve the problem. Tight upper bounds on the optimal objective value help to improve the performance. To find upper bounds on the optimal solution for the shortest longest path problem, we develop several heuristics and analyze their quality. On pearl graphs we prove a constant approximation ratio of 2 and 3 respectively for all heuristics. A numerical study on random pearl graphs shows that the solutions generated by the heuristics are usually much better than this worst-case bound. In Part II: "Construction and analysis of evacuation models using flows over time", we consider two optimization models in the context of evacuation planning. The first model is a parameter-based quickest flow model with time-dependent supply values. We give a detailed description of the network construction and of how different scenarios are modeled by scenario parameters. In a second step we analyze the effect of the scenario parameters on the evacuation time. Understanding how the different parameters influence the evacuation time allows us to provide better advice for evacuation planning and allows us to predict evacuation times without solving additional optimization problems. To understand the effect of the time-dependent supply values, we consider the quickest path problem with time-dependent supply values and provide a solution algorithm. The results from this consideration are generalized to approximate the behavior of the evacuation times in the context of quickest flow problems. The second model we consider is a path-based model for evacuation in the presence of a dynamic cost function. We discuss the challenges of this model and provide ideas for how to approach the problem from different angles. We relate the problem to the flow-decomposition problem and consider the computation of evacuation paths with dynamic costs for large capacities. For the latter method we provide heuristics to find paths and compare them to the optimal solutions by applying the methods to two evacuation scenarios. An analysis shows that the paths generated by the heuristic yield close to optimal solutions and in addition have several desirable properties for evacuation paths which are not given for the optimal solution.

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Author:Jan Peter Ohst
Referee:Stefan Ruzika, Stephan Westphal
Advisor:Stefan Ruzika
Document Type:Doctoral Thesis
Date of completion:2016/02/10
Date of publication:2016/02/11
Publishing institution:Universität Koblenz, Universitätsbibliothek
Granting institution:Universität Koblenz, Fachbereich 3
Date of final exam:2015/12/22
Release Date:2016/02/11
Tag:Evacuation modeling; Flow decomposition; Optimization; flows over time
Number of pages:174
Institutes:Fachbereich 3 / Mathematisches Institut
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):License LogoEs gilt das deutsche Urheberrecht: § 53 UrhG