An Exact Algorithm for Combining Vehicle Trips

  • Aristides Mingozzi
  • Lucio Bianco
  • Salvatore Ricciardelli
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 430)


We consider a vehicle scheduling problem arising in freight transport systems where a vehicle fleet of different classes, split among different depots, must be used to perform a set of trips over a time period horizon minimizing a given cost function. The set of trips must be partitioned in a number of disjoint sequences (called duties) and each duty must be assigned to a vehicle satisfying time window and vehicle-trip objection constraints. Moreover, a vehicle can perform only one duty and must return to the starting depot. This problem is an extension of a simpler problem known as Multiple Depot Vehicle Scheduling Problem (MD-VSP) that is NP hard.

In this paper we formulate the problem as Set Partitioning Problem with side constraints (SPP) where each column is a feasible vehicle duty. We describe a procedure for computing a lower bound to the optimal cost by finding an heuristic solution of the dual of the linear relaxation of the SPP formulation, without generating the entire SPP matrix. The dual solution obtained and an upper bound to the optimal solution cost are used to reduce the number of variables in the SPP in such a way that the resulting SPP can be solved by a branch and bound algorithm. The computational results show that the proposed method can be used to solve large size problems.


Feasible Solution Dual Solution Heuristic Solution Heuristic Procedure Crew Schedule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bertossi, A. A. / Carraresi, P. / Gallo, G. (1987): On some matching problems arising in vehicle scheduling models. in: Networks 17.Google Scholar
  2. Bianco, L. / Mingozzi, A. / Ricciardelli, S. (1994): A set partitioning approach to the multiple depot vehicle scheduling problem. in: Optimization Methods and Software 3.Google Scholar
  3. Bodin, L. / Golden, B. / Assad, A. / Ball, M. (1983): Routing and scheduling of vehicles and crews. in: Computers and Operations Research 10.Google Scholar
  4. Bodin, L. / Rosenfield, D. / Kydes, A. (1978): UCOST: A micro approach to a transit planning problem. in: Journal of Urban Analysis 5.Google Scholar
  5. Carpaneto, G. / Fischetti, M. / Toth, P. (1989): A Branch and Bound Algorithm for the Multiple Depot Vehicle Scheduling Problem. in: Networks 19.Google Scholar
  6. Dell’amico, M. / Fischetti, M. / Toth, P. (1990): Heuristic Algorithms for the Multiple Depot Vehicle Scheduling Problem. DEIS University of Bologna, Technical Report OR/90/3.Google Scholar
  7. Fisher, M. L. / Kedia, P. (1990): Optimal solution of set covering / partitioning problems using dual heuristics. in: Management Science 36.Google Scholar
  8. Marsten, R. E. / Shepardson, F. (1981): Exact solution of crew scheduling problems using the set partitioning model: Recent successfully applications. in: Networks 11.Google Scholar
  9. Mingozzi, A. / Ricciardelli, S. (1993): Heuristic procedures for the vehicle scheduling problem in freight transport system. Working Paper, Department of Mathematics, University of Bologna.Google Scholar
  10. Ribeiro, C.C. / Soumis, F. (1990): A column generation approach to the multiple depot vehicle scheduling problem. Research Report, GERAD, Montreal.Google Scholar
  11. Smith, B. / Wren, A. (1981): VAMPIRES and TASC: Two successfully applied bus scheduling programs. in: A. Wren (ed.): Computer scheduling of public transport: Urban passenger vehicle and crew scheduling. (North-Holland) AmsterdamGoogle Scholar
  12. Yen, J.Y. (1971): Finding the K-shortest loopless paths in a network. in: Management Science 17.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Aristides Mingozzi
    • 1
  • Lucio Bianco
    • 1
  • Salvatore Ricciardelli
    • 2
  1. 1.Department of MathematicsUniversity of BolognaBolognaItaly
  2. 2.Deparment of Electrical EngineeringUniversity “Tor Vergata”RomaItaly

Personalised recommendations