Michael R. Bussieck

Discrete optimization in public rail transport

Many problems arising in traffic planning can be modelled and solved using discrete optimization. We will focus on recent developments which were applied to large scale real world instances.Most railroad companies apply a hierarchically structured planning process. Starting with the definition of the underlying network used for transport one has to decide which infrastructural improvements are necessary. Usually, the rail system is periodically scheduled. A fundamental base of the schedule are the lines connecting several stations with a fixed frequency. Possible objectives for the construction of the line plan may be the minimization of the total cost or the maximization of the passengers’s comfort satisfying certain regulations. After the lines of the system are fixed, the train schedule can be determined. A criterion for the quality of a schedule is the total transit time of the passengers including the waiting time which should be minimized satisfying some operational constraints. For each trip of the schedule a train consisting of a locomotive and some carriages is needed for service. The assignment of rolling stock to schedule trips has to satisfy operational requirements. A comprehensible objective is to minimize the total cost. After all strategic and tactical planning the schedule has to be realized. Several external influences, for example delayed trains, force the dispatcher to recompute parts of the schedule on-line.

Optimal lines for railway systems

Abstract We discuss the optimal choice of traffic lines with periodic timetables on a railway system. A chosen line system has to offer sufficient capacity in order to serve the known amount of traffic on the system. The line optimization problem aims at the construction of a feasible line system optimizing certain objectives. We introduce a mixed integer linear programming formulation. For real world data we succeed in solving the model by means of suitable relaxations and sufficiently strong cutting planes with the commercial LP solver CPLEX 3.0.

The vertex set of a 0/1-polytope is strongly P -enumerable

Abstract In this paper, we discuss the computational complexity of the following enumeration problem: given a rational convex polyhedron P defined by a system of linear inequalities, output each vertex of P. It is still an open question whether there exists an algorithm for listing all vertices in running time polynomial in the input size and the output size. Informally speaking, a linear running time in the output size leads to the notion of P-enumerability introduced by Valiant (1979). The concept of strong P-enumerability additionally requires an output independent space complexity of the respective algorithm. We give such an algorithm for polytopes all of whose vertices are among the vertices of a polytope combinatorially equivalent to the hypercube. As a very important special case, this class of polytopes contains all 0 1 -polytopes. Our implementation based on the commercial LP solver CPLEX is superior to general vertex enumeration algorithms. We give an example how simplifications of our algorithm lead to efficient enumeration of combinatorial objects.

Scheduling trams in the morning

Abstract. In this note, we prove ??-completeness of the following problem: Given a set of trams of different types, which are stacked on sidings in their depot and an order in which trams of specified types are supposed to leave. Is there an assignment of trams to departure times without any shunting movements? In the particular case where the number of sidings is fixed, the problem is solvable in polynomial time. We derive a dynamic program and improve its performance by a state elimination scheme. We implemented three variants of the dynamic program and applied them to random data as well as to real-world data.

A fast algorithm for near cost optimal line plans

Abstract.We consider the design of line plans in public transport at a minimal total cost. Both, linear and nonlinear integer programming are adequate and intuitive modeling approaches for this problem. We present a heuristic variable fixing procedure which builds on problem knowledge from both techniques. We derive and compare lower bounds from different linearizations in order to assess the quality of our solutions. The involved integer linear programs are strengthened by means of problem specific valid inequalities. Computational results with practical data from the Dutch Railways indicate that our algorithm gives excellent solutions within minutes of computation time.

Optimal scrap combination for steel production

In steel production, scrap metal is used for cooling the enormous quantity of heat produced by blowing oxygen on hot metal. Scrap differs in regard to the content of iron and of some tramp elements. The price of the scrap depends on these attributes. Each melting bath unit of steel has its own material constraints for the amount of iron and tramp elements in order to guarantee the desired quality. In addition, the transportation of scrap is restricted because it needs time and space: the scrap is kept in some railroad cars in the scrap hall; empty cars must leave the hall, filled cars must be taken from several railroad tracks in the scrap yard and assembled to a train before transportation to the hall. There are upper limits for the number of cars in the hall and in the train, also for the number of railroad tracks used for assembly.Our objective is to find a minimum cost scrap combination for each melting bath unit of steel that obeys the material and transportation constraints. We model the problem using a MIP (mixed integer linear programming) approach. Real-life situations are solved with the commercial MIP-solver CPLEX. We present computational results which show significant improvement compared to the strategy applied today.ZusammenfassungIn der Stahlproduktion wird zur Kühlung des flüssigen Roheisens Metallschrott hinzugefügt. Dabei wird Schrott mit unterschiedlichem Gehalt an Eisen sowie an Spurenelementen eingesetzt. Abhängig von dieser Zusammensetzung variiert der Einkaufspreis für den Schrott. Für jeden produzierten Stahltyp sind gewisse Grenzwerte für Eisenanteil und den Gehalt an Spurenelementen im Stahl einzuhalten, um die geforderte Qualität zu erreichen. Der Schrott wird in Eisenbahnwaggons gelagert. Dadurch, daß Züge aus diesen Waggons gebildet werden müssen und diese Züge die Werkshalle auf dem vorgegebenen Gleisnetz erreichen bzw. verlassen müssen, entstehen zusätzlich zu den Materialrestriktionen auch noch Transportrestriktionen.Unser Ziel ist es, für jeden Produktionsprozeß die kostengünstigste Schrottzusammenstellung zu finden, so daß alle Material- und Transportrestriktionen eingehalten werden. Wir modellieren das Problem mit Hilfe eines gemischt-ganzzahligen linearen Programms (MIP) und lösen es mit dem kommerziellen MIP-Löser CPLEX. Unsere Rechenergebnisse für reale Produktionsserien zeigen bemerkenswerte Einsparungen gegenüber dem zur Zeit verwendeten Verfahren.

On balanced edge connectivity and applications to some bottleneck augmentation problems in networks

AbstractLetwi∶V ×V →Q,i = 1, 2 be two weight functions on the possible edges of a directed or undirected graph with vertex setV such that for the cut function, the inequality

A Fast Algorithm for Near Cost Optimal Line Plans

\delta _{w_2 } (T): = \sum\limits_{\scriptstyle i \in T \hfill \atop \scriptstyle j \notin T \hfill} {w_2 (ij) \ge 0.}