Evacuation dynamics influenced by spreading hazardous material

Simone Göttlich; Sebastian Kühn; Jan Peter Ohst; Stefan Ruzika; Markus Thiemann; Simone Göttlich; Sebastian Kühn; Jan Peter Ohst; Stefan Ruzika; Markus Thiemann

doi:10.3934/nhm.2011.6.443

Networks and Heterogeneous Media

2011, Volume 6, Issue 3: 443-464. doi: 10.3934/nhm.2011.6.443

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Evacuation dynamics influenced by spreading hazardous material

1.
Department of Mathematics, University of Mannheim, D-68131 Mannheim
2.
Department of Mathematics, TU Kaiserslautern, D-67663 Kaiserslautern

Received: 01 December 2010 Revised: 01 July 2011
90B10, 90B35.

In this article, an evacuation model describing the egress in case of danger is considered. The underlying evacuation model is based on continuous network flows, while the spread of some gaseous hazardous material relies on an advection-diffusion equation. The contribution of this work is twofold. First, we introduce a continuous model coupled to the propagation of hazardous material where special cost functions allow for incorporating the predicted spread into an optimal planning of the egress. Optimality can thereby be understood with respect to two different measures: fastest egress and safest evacuation. Since this modeling approach leads to a pde/ode-restricted optimization problem, the continuous model is transferred into a discrete network flow model under some linearity assumptions. Second, it is demonstrated that this reformulation results in an efficient algorithm always leading to the global optimum. A computational case study shows benefits and drawbacks of the models for different evacuation scenarios.
- Evacuation,
- dynamic network flows,
- optimization
Citation: Simone Göttlich, Sebastian Kühn, Jan Peter Ohst, Stefan Ruzika, Markus Thiemann. Evacuation dynamics influenced by spreading hazardous material[J]. Networks and Heterogeneous Media, 2011, 6(3): 443-464. doi: 10.3934/nhm.2011.6.443

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Abstract

In this article, an evacuation model describing the egress in case of danger is considered. The underlying evacuation model is based on continuous network flows, while the spread of some gaseous hazardous material relies on an advection-diffusion equation. The contribution of this work is twofold. First, we introduce a continuous model coupled to the propagation of hazardous material where special cost functions allow for incorporating the predicted spread into an optimal planning of the egress. Optimality can thereby be understood with respect to two different measures: fastest egress and safest evacuation. Since this modeling approach leads to a pde/ode-restricted optimization problem, the continuous model is transferred into a discrete network flow model under some linearity assumptions. Second, it is demonstrated that this reformulation results in an efficient algorithm always leading to the global optimum. A computational case study shows benefits and drawbacks of the models for different evacuation scenarios.

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