Modeling and optimizing dynamic dial-a-ride problems Alberto Colorni and Giovanni Righini a Dipartimento di Elettronica e Informazione, Politecnico di Milano, Italy E-mail: colorni@elet.polimi.it a Dipartimento di Scienze dell'Informazione, Polo Didattico e di Ricerca di Crema, Universita Á degli Studi di Milano, Italy E-mail: righini@crema.unimi.it Received 24 June 1999; received in revised form 30 June 2000; accepted 28 August 2000 Abstract Dial-a-ride is an emerging alternative to traditional public transportation systems. The aim of this paper is to reduce the gap between the models studied in optimization literature and the requirements of practical applications. We also describe the algorithms implemented in DARIA, a PC program for the optimization of static and dynamic dial-a-ride problems. We brie¯y illustrate two case studies and future developments of the DARIA project. Keywords: transportation, dial-a-ride, branch-and-bound, local search 1. Introduction In modern cities transportation systems are intensively exploited by a number of people. Their ef®ciency and comfort become more and more important, together with the need for making public- transportation means competitive and appealing, in order to reduce traf®c, noise, pollution, and delays (Uchimura, Saitoh and Takahashi, 1999). For these reasons, transportation-systems planners and managers and public administrators show an increasing interest in the dial-a-ride option (Colorni, 1999). It offers the comfort and ¯exibility of private cars and taxis at lower cost, through a better exploitation of vehicles' capacity. Public mass-transport systems based on trains, underground, buses, and tramways are cheaper than dial-a-ride, but far less comfortable because the vehicles follow ®xed routes and obey ®xed time-schedules. Dial-a-ride systems can also be integrated with traditional ones, yielding multi-modal transportation systems (Liaw, White and Bander, 1996). The name `dial-a-ride' comes from the phone call by which a customer is supposed to ask for service. A central planning unit receives all customer calls and decides the allocations of the customers to the vehicles, and the route of each vehicle. The combinatorial problem of deciding how to allocate customers to vehicles and how to route each vehicle is hard from the viewpoint of computational complexity theory, since it is a generalization and a combination of the set partitioning problem and the minimum Hamiltonian path Intl. Trans. in Op. Res. 8 (2001) 155±166 # International Federation of Operational Research Societies. Published by Blackwell Publishers Ltd.