Uwe T. Zimmermann

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.

Real-time dispatch of trams in storage yards

Real-time dispatch problems arise when preparing and executing the daily schedule of local transport companies. We consider the daily dispatch of transport vehicles like trams in storage yards. Immediately on arrival, each tram has to be assigned to a location in the depot and to an appropriate round trip of the next schedule period. In order to achieve a departure order satisfying the scheduled demand, shunting of vehicles may be unavoidable. Since shunting takes time and causes operational cost, the number of shunting movements should be minimized without violation of operational constraints. As an alternative, we may serve some round trips with trams of type differing from the requested type.In practice, the actual arrival order of trams may differ substantially from the scheduled arrival order. Then, dispatch decisions are due within a short time interval and have to be based on incomplete information. For such real-time dispatch problems, we develop combinatorial optimization models and exact as well as heuristic algorithms. Computational experience for real-world and random data shows that the derived methods yield good (often optimal) solutions within the required tight time bounds.

Cost optimal periodic train scheduling

For real world railroad networks, we consider minimizing operational cost of train schedules which depend on choosing different train types of diverse speed and cost. We develop a mixed integer linear programming model for this train scheduling problem. For practical problem sizes, it seems to be impossible to directly solve the model within a reasonable amount of time. However, suitable decomposition leads to much better performance. In the first part of the decomposition, only the train type related constraints stay active. In the second part, using an optimal solution of this relaxation, we select and fix train types and try to generate a train schedule satisfying the remaining constraints. This decomposition idea provides the cornerstone for an algorithm integrating cutting planes and branch-and-bound. We present computational results for railroad networks from Germany and the Netherlands.

Optimal Sorting of Rolling Stock at Hump Yards

In this paper we provide a quite general description of a class of problems called Sorting of Rolling Stock Problem(s). An SRSP consists in finding an optimal schedule for rearranging units of rolling stock (railcars, trams, trains, . . . ) at shunting yards, covering a broad range of specially structured applications. Here, we focus on versions of SRSP at particular shunting yards featuring a hump. We analyze the use of such a hump yard in our research project Zeitkritische Ablaufbergoptimierung in Rangierbahnhofen 1 in cooperation with BASF, The Chemical Company, in Ludwigshafen. Among other results we present a remarkably efficient algorithm with linear running time for solving the practical SRSP at the BASF hump yard.

Engine Routing and Scheduling at Industrial In-Plant Railroads

In-plant railroad engine scheduling involves routing and scheduling decisions for a heterogeneous fleet of switching engines to serve a set of time-window- and capacity-constrained transportation requests. Despite an ever-increasing competition, the current planning is purely by pencil and paper. Our paper describes the mathematical and algorithmic developments for addressing in-plant railroad decision support for scheduling and routing. The problem discussed in our work is related to the multiple-vehicle pickup and delivery problem. Exploiting the structure of admissible schedules of our particular railroad situation, we introduce two formulations of the problem as mixed integer and set partitioning programs. We propose solving the linear programming relaxation of the set partition model by column generation. We focus on the pricing problem stated in the form of a constrained shortest path problem, which isNP complete in the strong sense. A new exact label correcting algorithm is developed that prunes the search space in a novel manner. Heuristically obtained integer solutions of a practical quality are proposed as well. All the claims are demonstrated by computational experiments on both artificial and real-life data. We discuss implementation details as well.

Combinatorial Online Optimization in Real Time

Optimization is the task of finding a best solution to a given problem. When the decision variables are discrete we speak of a combinatorial optimization problem. Such a problem is online when decisions have to be made before all data of the problem are known. And we speak of a real-time online problem when online decisions have to be computed within very tight time bounds. This paper surveys the art of combinatorial online and realtime optimization, it discusses, in particular, the concepts with which online and real-time algorithms can be analyzed.

A strongly polynomial algorithm for minimum cost submodular flow problems

The only known strongly polynomial algorithm for solving minimum cost submodular flow problems is due to Frank and Tardos Frank, A., E. Tardos. 1985. An application of the simultaneous approximation in combinatorial optimization. Report No. 85375, Institut fur Okonometrie und Operations Research, Bonn, May. and is based on the simultaneous approximation algorithm of Lenstra, Lenstra, and Lovasz Lenstra, A. K., H. W. Lenstra, L. Lovasz. 1982. Factoring polynomials with rational coefficients. Math. Ann.261 515--534.. We propose a purely combinatorial strongly polynomial algorithm. It consists in solving a sequence of at most m + nn-1 minimum cost submodular flow problems with cost coefficients bounded by n2, where n is the number of the vertices and m is the number of the arcs in the underlying graph. The current cost coefficients are calculated by means of tree projection and scaling.

Stowage and Transport Optimization in Ship Planning

We consider the ship planning problem at maritime container terminals where containers are loaded onto and discharged from ships using quay cranes. The container transport between the ships and the yard positions in the terminal is carried out by a fleet of straddle carriers. Based on a stowage plan provided by the shipping company, the dispatcher assigns containers to specified bay positions. Then, subject to operational and stability constraints, he schedules containers in order to avoid waiting times at the quay cranes. We propose an approach combining stowage planning and the selection of “good” loading and transport sequences. For a just-in-time scheduling model, we present computational results based on real-world data of a German container terminal. Moreover, we discuss some real-time and online influences on the daily dispatch situation.

A combinatorial interior point method for network flow problems

For solving minimum cost flow problems, we develop a combinatorial interior point method based on a variant of the algorithm of Karmarkar, described in Gonzaga [3, 4]. Gonzaga proposes search directions generated by projecting certain directions onto the nullspace ofA. By the special combinatorial structure of networks any projection onto the nullspace ofA can be interpreted as a flow in the incremental graph ofG. In particular, to evaluate the new search direction, it is sufficient to choose a negative circuit subject to costs on the arcs depending on the current solution. That approach results in an O(mn2L) algorithm wherem denotes the number of vertices,n denotes the number of arcs, andL denotes the total length of the input data.