Algorithmenbasierte Produktentwicklung für integrale Blechbauweisen höherer Verzweigungsordnung - Wäldele, Martin; Fügenschuh, Armin-René; Birkhofer, Herbert; Martin, Alexander (5. Gemeinsames Kolloquium Konstruktionstechnik 2007, 2007)

A Multicriterial Approach for Optimizing Bus Schedules and School Starting Times - Fügenschuh, Armin-René; Martin, Alexander (Annals of Operations Research, 2006)
In many rural counties pupils on their way to school are a large, if not the largest group of customers for public mass transit. Hence an effective optimization of public mass transit in these regions must include the traffic caused by pupils. Besides a change in the schedules of the buses and the starting times of the trips, the school starting time may become an integral part of the planning process. We discuss the legal framework for this optimization problem in German states and counties and present a multi-objective mixed-integer linear programming formulation for the simultaneous specification of school and trip starting times. For its solution, we develop a two stage decomposition heuristic and apply it to practical data sets from three different rural German counties.
Keywords: Mixed-integer programming; Multicriteria optimization; Multiple traveling salesman problem; Time windows

Combinatorial and Continuous Models for the Optimization of Traffic Flows on Networks - Fügenschuh, Armin-René; Herty, Michael; Klar, Axel; Martin, Alexander (SIAM Journal on Optimization, 2006)
A hierachy of simplified models for traffic flow on networks is derived from continuous traffic flow models based on partial differential equations. The hierachy contains nonlinear and linear combinatorial models with and without dynamics. Optimization problems are treated for all models and numerical results and algorithms are compared.
Keywords: Combinatorial methods; Partial differential equations; Traffic networks

Computational Integer Programming and Cutting Planes - Fügenschuh, Armin-René; Martin, Alexander (Handbooks in Operations Research and Management Science, 2005)
The study and solution of mixed-integer programming problems is of great interest, because they arise in a variety of mathematical and practical applications. Today's state-of-art software packages for solving mixed-integer programs based on linear programming include preprocessing, branch-and-bound, and cutting planes techniques. The main purpose of this article is to describe these components and recent developments that can be found in many solvers. Besides linear programming based relaxation methods we also discuss Langrangean, Dantzig-Wolfe and Benders' decomposition and their interrelations.

Diskrete und kontinuierliche Modelle zur Topologie- und Geometrie-Optimierung von Blechprofilen - Fügenschuh, Armin-René; Hess, Wolfgang; Martin, Alexander; Ulbrich, Stefan (Tagungsband 1. Zwischenkolloqium SFB 666, 2007)

Efficient Reformulation and Solution of a Nonlinear PDE-Controlled Flow Network Model - Fügenschuh, Armin-René; Goettlich, S.; Herty, Michael; Kirchner, C.; Martin, Alexander (Computing, 2009)
We consider a flow network where the flow of parts can be controlled at the vertices of the network. Based on a modified coarse grid discretization presented in Fügenschuh et al. (SIAM J Scientific Comput 30(3):1490-1507, 2008) we derive a mixed-integer program (MIP). Under suitable assumptions on the cost functional we prove that there exists an equivalent linear program (LP). We present numerical results concerning validity of our result and show the improvement of the computing times using the equivalent LP over the MIP.
Keywords: Adjoint calculus; Conservation laws; Flow networks; Networks

Integrated Optimization of School Starting Times and Public Bus Services - Fügenschuh, Armin-René; Martin, Alexander; et al. (Operations Research Proceedings 2004, 2005)
In many rural areas, the public bus service is demand-oriented: By far the biggest group of customers are pupils who are transported to their schools within certain strict time limits. Usually, all schools start around the same time, which causes a morning peak in the number of deployed buses. However, schools are allowed to change their starting times within some interval. The question is, how to simultanenously rectify the starting times for all schools and bus trips in a certain county so that the number of scheduled buses is minimal. This problem can be formulated as a vehicle routing problem with coupled time windows (VRP-CTW), which is an extension of the vehicle routing problem with time windows (VRP-TW), where additional coupling constraints on the time windows are introduced. We give a mixed-integer programming formulation for VRP-CTW, and present solutions and lower bounds for randomly generated and real-world instances.

Mixed-Integer Models for Topology Optimization in Sheet Metal Design - Fügenschuh, Armin-René; Martin, Alexander (PAMM, Proceedings of Applied Mathematics and Mechanics, 2007)

The coolest path problem - Frank, Martin; Fügenschuh, Armin-René; Herty, Michael; Schewe, Lars (Networks and Heterogeneous Media, 2010)
We introduce the coolest path problem, which is a mixture of two well-known problems from distinct mathematical fields. One of them is the shortest path problem from combinatorial optimization. The other is the heat conduction problem from the field of partial differential equations. Together, they make up a control problem, where some geometrical object traverses a digraph in an optimal way, with constraints on intermediate or the final state. We discuss some properties of the problem and present numerical solution techniques. We demonstrate that the problem can be formulated as a linear mixed-integer program. Numerical solutions can thus be achieved within one hour for instances with up to 70 nodes in the graph. © American Institute of Mathematical Sciences.
Keywords: Heat equation; Integer programming; Shortest path

The Transport PDE and Mixed-Integer Linear Programming - Fügenschuh, Armin-René; Geißler, Björn; Martin, Alexander; Morsi, Antonio (Models and Algorithms for Optimization in Logistics, 2009)
Discrete, nonlinear and PDE constrained optimization are mostly considered as different fields of mathematical research. Nevertheless many real-life problems are most naturally modeled as PDE constrained mixed integer nonlinear programs. For example, nonlinear network flow problems where the flow dynamics are governed by a transport equation are of this type. We present four different applications together with the derivation of the associated transport equations and we show how to model these problems in terms of mixed integer linear constraints.
Keywords: Transport Equation; Partial Differential Equation; Mixed-Integer Linear Programming; Modeling; Nonlinear Constraints

Verfeinerte Modelle zur Topologie- und Geometrie-Optimierung von Blechprofilen mit Kammern - Fügenschuh, Armin-René; Hess, Wolfgang; Schewe, Lars; Martin, Alexander; Ulbrich, Stefan (Tagungsband 2. Zwischenkolloqium SFB 666, 2008)