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The steel mill slab design problem from the CSPLIB is a combinatorial
optimization problem motivated by an application of the steel industry. It
has been widely studied in the constraint programming community. Several
methods were proposed to solve this problem. A steel mill slab library was
created which contains 380 instances. A closely related binpacking problem
called the multiple knapsack problem with color constraints, originated
from the same industrial problem, was discussed in the integer programming
community. In particular, a simple integer program for this problem has
been given by Forrest et al. The aim of this paper is to bring these
different studies together. Moreover, we adapt the model of Forrest et
al. for the steel mill slab design problem. Using this model and a
state-of-the-art integer program solver all instances of the steel mill
slab library can be solved efficiently to optimality. We improved,
thereby, the solution values of 76 instances compared to previous results.
Finally, we consider a recently introduced variant of the steel mill slab
design problem, where within all solutions which minimize the leftover one
is interested in a solution which requires a minimum number of slabs. For
that variant we introduce two approaches and solve all instances of the
steel mill slab library with this slightly changed objective function to
optimality.

We consider an auction of slots to run trains through a railway network. In contrast to the classical setting for combinatorial auctions, there is not only competition for slots, but slots can mutually exclude each other, such that general conflict constraints on bids arise. This turns the winner determination problem associated with such an auction into a complex combinatorial optimization problem. It also raises a number of auction design questions, in particular, on incentive compatibilty. We propose a single-shot second price auction for railway slots, the Vickrey Track Auction (VTA). We show that this auction is incentive compatible, i.e., rational bidders are always motivated to bid their true valuation, and that it produces efficient allocations, even in the presence of constraints on allocations. These properties are, however, lost when rules on the submission of bids such as, e.g., lowest bids, are imposed. Our results carry over to generalized" Vickrey auctions with combinatorial constraints.

Railway scheduling is based on the principle of the construction of a conflict-free timetable. This leads to a strict definition of capacity: in contrast with road transportation, it can be said in advance whether a given railway infrastructure can accommodate - at least in theory - a certain set of train requests. Consequently, auctions for railway capacity are modeled as auctions of discrete goods -- the train slots. We present estimates for the efficiency gain that may be generated by slot auctioning in comparison with list price allocation. We introduce a new class of allocation and auction problems, the feasible assignment problem, that is a proper generalization of the well-known combinatorial auction problem. The feasible assignment class was designed to cover the needs for an auction mechanism for railway slot auctions, but is of interest in its own right. As a practical instance to state and solve the railway slot allocation problem, we present an integer programming formulation, briefly the ACP, which turns out to be an instance of the feasible assignment problem and whose dual problem yields prices that can be applied to define a useful activity rule for the linearized version of the Ausubel Milgrom Proxy auction. We perform a simulation aiming to measure the impact on efficiency and convergence rate.

The steel mill slab design problem from the CSPLib is a binpacking problem that is motivated by an application of the steel industry and that has been widely studied in the constraint programming community. Recently, several people proposed new models and methods to solve this problem. A steel mill slab library was created which contains 380 instances. A closely related binpacking problem called multiple knapsack problem with color constraints, originated from the same industrial problem, were discussed in the integer programming community. In particular, a simple integer programming for this problem has been given by Forrest et al. [3]. The aim of this paper is to bring these different studies together. Moreover, we adopt the model of [3] for the steel mill slab problem. Using a state of the art integer program solver, this model is capable to solve all instances of the steel mill slab library, mostly in less than one second, to optimality. We improved, thereby, the solution value of 76 instances.

The track allocation problem, also known as train routing problem or train timetabling problem, is to find a conflict-free set of train routes of maximum value in a railway network. Although it can be modeled as a standard path packing problem, instances of sizes relevant for real-world railway applications could not be solved up to now. We propose a rapid branching column generation approach that integrates the solution of the LP relaxation of a path coupling formulation of the problem with a special rounding heuristic. The approach is based on and exploits special properties of the bundle method for the approximate solution of convex piecewise linear functions. Computational results for difficult instances of the benchmark library TTPLIB are reported.

In this paper a bottom-up approach of automatic simplification of a railway network is presented. Starting from a very detailed, microscopic level, as it is used in railway simulation, the network is transformed by an algorithm to a less detailed level (macroscopic network), that is sufficient for long-term planning and optimization. In addition running and headway times are rounded to a pre-chosen time discretization by a special cumulative method, which we will present and analyse in this paper. After the transformation we fill the network with given train requests to compute an optimal slot allocation. Then the optimized schedule is re-transformed into the microscopic level and can be simulated without any conflicts occuring between the slots. The algorithm is used to transform the network of the very dense Simplon corridor between Swiss and Italy. With our aggregation it is possible for the first time to generate a profit maximal and conflict free timetable for the corridor across a day by a simultaneously optimization run.

This paper presents a case study on a railway timetable optimization for the very dense Simplon corridor, a major railway connection in the Alps between Switzerland and Italy. Starting from a detailed microscopic network as it is used in railway simulation, the data is transformed by an automatic procedure to a less detailed macroscopic network, that is sufficient for the purpose of capacity planning and amenable to state-of-the-art integer programming optimization methods. In this way, the macroscopic railway network is saturated with trains. Finally, the corresponding timetable is re-transformed to the microscopic level in such a way that it can be operated without any conflicts among the slots. Using this integer programming based micro-macro aggregation-disaggregation approach, it becomes for the first time possible to generate a profit maximal and conflict free timetable for the complete Simplon corridor over an entire day by a simultaneous optimization of all trains requests. This also allows to to undertake a sensitivity analysis of various problem parameters.

The airline crew scheduling problem deals with the construction of crew rotations in order to cover the flights of a given schedule at minimum cost. The problem involves complex rules for the legality and costs of individual pairings and base constraints for the availability of crews at home bases. A typical instance considers a planning horizon of one month and several thousand flights. We propose a column generation approach for solving airline crew scheduling problems that is based on a set partitioning model. We discuss algorithmic aspects such as the use of bundle techniques for the fast, approximate solution of linear programs, a pairing generator that combines Lagrangean shortest path and callback techniques, and a novel rapid branching'' IP heuristic. Computational results for a number of industrial instances are reported. Our approach has been implemented within the commercial crew scheduling system NetLine/Crew of Lufthansa Systems Berlin GmbH.

We present an approach to implement an auction of railway slots. Railway network, train driving characteristics, and safety requirements are described by a simplified, but still complex macroscopic model. In this environment, slots are modelled as combinations of scheduled track segments. The auction design builds on the iterative combinatorial auction. However, combinatorial bids are restricted to some types of slot bundles that realize positive synergies between slots. We present a bidding language that allows bidding for these slot bundles. An integer programming approach is proposed to solve the winner determination problem of our auction. Computational results for auction simulations in the Hannover-Fulda-Kassel area of the German railway network give evidence that auction approaches can induce a more efficient use of railway capacity.

This article is about the optimal track allocation problem (OPTRA) to find, in a given railway network, a conflict free set of train routes of maximum value. We study two types of integer programming formulations: a standard formulation that models block conflicts in terms of packing constraints, and a new extended formulation that is based on additional configuration' variables. We show that the packing constraints in the standard formulation stem from an interval graph, and that they can be separated in polynomial time. It follows that the LP relaxation of a strong version of this model, including all clique inequalities from block conflicts, can be solved in polynomial time. We prove that the extended formulation produces the same LP bound, and that it can also be computed with this model in polynomial time. Albeit the two formulations are in this sense equivalent, the extended formulation has advantages from a computational point of view, because it features a constant number of rows and is therefore amenable to standard column generation techniques. Results of an empirical model comparison on mesoscopic data for the Hannover-Fulda-Kassel region of the German long distance railway network are reported.