Line Planning on Paths and Tree Networks withAppli ations to the Quito Trolebús

Appli ations to the Quito Trolebús System Luis M. Torres1, Ramiro Torres1, Ralf Borndörfer2, and Mar E. Pfets h2 1 Es uela Polité ni a Na ional, Quito, E uador, {ltorres,rtorres} math.epn.edu.e 2 Zuse Institute Berlin, 14195 Berlin, Germany, {borndoerfer,pfets h} zib.de Abstra t. Line planning is an important step in the strategi planning pro ess of a publi transportation system. In this paper, we dis uss an optimization model for this problem in order to minimize operation osts while guaranteeing a ertain level of quality of servi e, in terms of available transport apa ity. We analyze the problem for path and tree network topologies as well as several ategories of line operation that are important for the Quito Trolebús system. It turns out that, from a omputational omplexity worst ase point of view, the problem is hard in all but the most simple variants. In pra ti e, however, instan es based on real data from the Trolebús System in Quito an be solved quite well, and signi ant optimization potentials an be demonstrated. 1 Introdu tion The major ities of South Ameri a are fa ing an enormous and onstantly inreasing demand for transportation and, unfortunately, also in rease vehi ular ongestion, with all its negative e e ts. In Quito, the elongated topography of the ity with 1.8 millions inhabitants (the urban area being 60 km long and 8 km wide) aggravates vehi ular ongestion even more, su h that tra almost ompletely breaks down during rush hours. As a onsequen e, the lo al government fa es the ne essity of improving the publi mass transit system. A lowost option that has produ ed satisfa tory results in re ent years has been the implementation of major orridors of transportation. These orridors onsist of street tra ks that are reserved ex lusively for highapa ity bus units, whi h, in this way, an operate independently of the rest of the tra . Even though the topology of a orridor is extremely simple (just a path), bus operation on it is non-trivial. In fa t, it is usually organized in a omplex system of several dozen lines, whi h over, in an overlapping way, di erent parts of the orridor, and whi h an operate in di erent ways, e.g., as normal lines or as express lines (stopping only at distinguished express stations), as open lines (unidire tional) or losed lines (bidire tional lines), and in any ombination of these ategories. The orridor lines are often omplemented by feeding lines that transport passengers between spe ial transshipment terminals of the orridor and the nearby neighborhoods.