T he line-planning problem is one of the fundamental problems in strategic planning of public and rail transport. It involves finding lines and corresponding frequencies in a transport network such that a given travel demand can be satisfied. There are (at least) two objectives: the transport company wishes to minimize operating costs, and the passengers want to minimize traveling times. We propose a new multicommodity flow model for line planning. Its main features, in comparison to existing models, are that the passenger paths can be freely routed and lines are generated dynamically. We discuss properties of this model, investigate its complexity, and present a column-generation algorithm for its solution. Computational results with data for the city of Potsdam, Germany, are reported.
In this paper we investigate whether matrices arising from linear or integer programming problems can be decomposed into so-called bordered block diagonal form. More precisely, given some matrix A, we try to assign as many rows as possible to some number ß of blocks of size K such that no two rows assigned to different blocks intersect in a common column. Bordered block diagonal form is desirable because it can guide and speed up the solution process for linear and integer programming problems. We show that various matrices from the LP-and MIP-libraries Netlib and Miplib can indeed be decomposed into this form by computing optimal decompositions or decompositions with proven quality These computations are done with a branch-and-cut algorithm based on polyhedral investigations of the atrix decomposition problem. In practice, however, one would use heuristics to find a good decomposition. We present several heuristic ideas and test their erformance. Finally, we investigate the usefulness of optimal matrix decompositions into bordered block diagonal form for integer programming by using such decompositions to guide the branching process in a branch-and-cut code for general mixed integer programs Keywords, block structure of a sparse matrix, matrix decomosition, integer programming, polyhedral combinatorics, cutting planes athematics Subjct Classification (MSC 1991) 90C10, 65F50
Abstract:Telebus is Berlin's dial-a-ride system for handicapped people who cannot use the public transportation system. The service is provided by a fleet of about 100 mini-buses and includes assistance in getting in and out of the vehicle. Telebus has between 1,000 and 1,500 transportation requests per day. The problem is to schedule these requests onto the vehicles such that punctual service is provided while operation costs are minimized. Addi tional constraints include pre-rented vehicles, fixed bus driver shift lengths, obligatory breaks, and different vehicle capacities.We use a set partitioning approach for the solution of the bus schedul ing problem that consists of two steps. The first clustering step identifies segments of possible bus tours ("orders") such that more than one person is transported at a time; the aim in this step is to reduce the size of the problem and to make use of larger vehicle capacities. The problem of selecting a set of orders such that the traveling distance of the vehicles within the orders is minimal is a set partitioning problem that can be solved to optimality. In the second step the selected orders are chained to yield possible bus tours respecting all side constraints. The problem to select a set of bus tours such that each order is serviced once and such that the total traveling distance of the vehicles is minimum is again a set partitioning problem that is solved approximately.We have developed a computer system for the solution of the bus schedul ing problem that includes a branch-and-cut algorithm for the solution of the set partitioning problems. A version of this system has been in operation at Telebus since July 1995. Its use made it possible for Telebus to serve about 30% more requests per day with the same resources.
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