Quickest Cluster Flow Problems
Macroscopic models based on dynamic network flow theory are successfully applied to obtain lower bounds on real evacuation times . The goal of our research is to tighten this lower bound and to make this macroscopic approach more realistic by taking into account clustering of evacuees - a sociological phenomenon observed in evacuation scenarios. A cluster of flow units in the network flow model represents families or cliques which tend not to move independently but as groups . This fact is not covered by macroscopic approaches based on classical network flow theory. In this article, we take clustering into account and thus improve existing macroscopic network flow models. We focus on two different sizes of groups traversing the network, modeled as single flow units and cluster flow units the latter of which occupy d times as much capacity as single flow units. In this novel approach, we are given fixed amounts of single flow units and cluster flow units and minimize the time at which the last (single or cluster) unit reaches the target. We present an algorithm that gives a 2-approximation for general networks and is optimal for the subclass of series-parallel networks.
KeywordsFlow Problem Flow Unit Macroscopic Approach Optimal Makespan Cluster Flow
Unable to display preview. Download preview PDF.
This paper is supported in part by the Federal Ministry for Education and Research (Bundesministerium für Bildung und Forschung, BMBF), Project REPKA, under FKZ 13N9961 (TU KL). We thank Heike Sperber, Technical University of Kaiserslautern, for several fruitful discussions on flows in series-parallel networks.
- 1.Hamacher, H.W., Tjandra, S.A.: Mathematical Modelling of Evacuation Problems – A State of the Art. Pedestrian and Evacuation Dynamics. 227-266 (2002)Google Scholar
- 3.Hamacher, H.W., Heller, S., Klein, W., Köster, G., Ruzika, S.: A Sandwich Approach for Evacuation Time Bounds. To appear in conference proceedings PED (2010)Google Scholar
- 4.Kneidl, A., Thiemann, M., Borrmann, A., Ruzika, S., Hamacher, H.W., Köster, G., Rank, E.: Bidirectional Coupling of Macroscopic and Microscopic Approaches for Pedestrian Behavior Prediction. To appear in conference proceedings PED (2010)Google Scholar
- 9.Tjandra, S.A.: Dynamic Network Optimization with Application to the Evacuation Problem. Ph.D. Thesis, University of Kaiserslautern (2003)Google Scholar
- 10.Valdes, J., Tarjan, R.E., Lawler, E.L.: The recognition of series parallel digraphs. Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing. ACM (1979)Google Scholar
- 11.Ruzika, S., Sperber, H., Steiner, M.: Earliest Arrival Flows on Series-Parallel Graphs. Report in Wirtschaftsmathematik 122. TU Kaiserslautern (2009)Google Scholar