Simone Göttlich

Optimal Control for Continuous Supply Network Models

We consider a supply network where the flow of parts can be controlled at the vertices of the network. Based on a coarse grid discretization provided in [6] we derive discrete adjoint equations which are subsequently validated by the continuous adjoint calculus. Moreover, we present numerical results concerning the quality of approximations and computing times of the presented approaches.

A Discrete Optimization Approach to Large Scale Supply Networks Based on Partial Differential Equations

We introduce a continuous optimal control problem governed by ordinary and partial differential equations for supply chains on networks. We derive a mixed-integer model by discretization of the dynamics of the partial differential equations and by approximations to the cost functional. Finally, we investigate numerically properties of the derived mixed-integer model and present numerical results for a real-world example.

THE SCALAR KELLER–SEGEL MODEL ON NETWORKS

In this work, we extend the one-dimensional Keller–Segel model for chemotaxis to general network topologies. We define appropriate coupling conditions ensuring the conservation of mass and show the existence and uniqueness of the solution. For our computational studies, we use a positive preserving first-order scheme satisfying a network CFL condition. Finally, we numerically validate the Keller–Segel network model and present results regarding special network geometries.

Optimization of order policies in supply networks

The purpose of this paper is to develop a model which allows for the study and optimization of arbitrarily complex supply networks, including order policies and money flows. We propose a mathematical description that captures the dynamic behavior of the system by a coupled system of ordinary differential delay equations. The underlying optimization problem is solved using discretization techniques yielding a mixed-integer programming problem.

Speed limit and ramp meter control for traffic flow networks

The control of traffic flow can be related to different applications. In this work, a method to manage variable speed limits combined with coordinated ramp metering within the framework of the Lighthill–Whitham–Richards (LWR) network model is introduced. Following a ‘first-discretize-then-optimize’ approach, the first order optimality system is derived and the switch of speeds at certain fixed points in time is explained, together with the boundary control for the ramp metering. Sequential quadratic programming methods are used to solve the control problem numerically. For application purposes, experimental setups are presented wherein variable speed limits are used as a traffic guidance system to avoid traffic jams on highway interchanges and on-ramps.

Particle methods for pedestrian flow models: From microscopic to nonlocal continuum models

A hierarchy of models for pedestrian flow is numerically investigated using particle methods. It includes microscopic models based on interacting particle system coupled to an eikonal equation, hydrodynamic models using equations for density and mean velocity, nonlocal continuum equations for the density and diffusive Hughes equations. Particle methods are used on all levels of the hierarchy. Numerical test cases are investigated by comparing the above models.

Modeling and optimizing traffic light settings in road networks

We discuss continuous traffic flow network models including traffic lights. A mathematical model for traffic light settings within a macroscopic continuous traffic flow network is presented, and theoretical properties are investigated. The switching of the traffic light states is modeled as a discrete decision and is subject to optimization. A numerical approach for the optimization of switching points as a function of time based upon the macroscopic traffic flow model is proposed. The numerical discussion relies on an equivalent reformulation of the original problem as well as a mixed-integer discretization of the flow dynamics. The large-scale optimization problem is solved using derived heuristics within the optimization process. Numerical experiments are presented for a single intersection as well as for a road network.

A Network Model for Supply Chains with Multiple Policies

In the present paper a network model for supply chains with policy attributes is introduced. The proposed network model is an extension of the single lane model with policy attributes presented in [D. Armbruster, P. Degond, and C. Ringhofer, Transport Theory Statist. Phys., submitted]. The single lane model is extended to the network case using the ideas developed in [S. Gottlich, M. Herty, and A. Klar, Commun. Math. Sci., 3 (2005), pp. 545–559; S. Gottlich, M. Herty, and A. Klar, Commun. Math. Sci., 4 (2006), pp. 315–330]. Numerical results are presented for several different examples.

Evacuation dynamics influenced by spreading hazardous material

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.

A scalar conservation law with discontinuous flux for supply chains with finite buffers

An aggregate continuum model for production flows and supply chains with fi- nite buffers is proposed and analyzed. The model extends earlier partial differential equations that represent deterministic coarse grained models of stochastic production systems based on mass con- servation. The finite size buffers lead to a discontinuous clearing function describing the throughput as a function of the work in progress (WIP). Following previous work on stationary distribution of WIP along the production line, the clearing function becomes dependent on the production stage and decays linearly as a function of the distance from the end of the production line. A transient experi- ment representing the breakdown of the last machine in the production line and its subsequent repair is analyzed analytically and numerically. Shock waves and rarefaction waves generated by blocking and reopening of the production line are determined. It is shown that the time to shutdown of the complete flow line is much shorter than the time to recovery from a shutdown. The former evolves on a transportation time scale, whereas the latter evolves on a much longer time scale. Comparisons with discrete event simulations of the same experiment are made.