A Binary Quadratic Programming Approach to the Vehicle Positioning Problem
The Vehicle Positioning Problem (VPP) is a classical combinatorial optimization problem that has a natural formulation as a Mixed Integer Quadratically Constrained Program. This MIQCP is closely related to the Quadratic Assignment Problem and, as far as we know, has not received any attention yet. We show in this article that such a formulation has interesting theoretical properties. Its QP relaxation produces, in particular, the first known nontrivial lower bound on the number of shuntings. In our experiments, it also outperformed alternative integer linear models computationally. The strengthening technique that raises the lower bound might also be useful for other combinatorial optimization problems.
KeywordsQuadratic Assignment Problem Mixed Integer Nonlinear Programming Integer Quadratic Programming Parking Position Integer Linear Model
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- 1.T. Achterberg, Constraint integer programming, Ph.D. thesis, TU Berlin, (2007).Google Scholar
- 2.A. Billionnet and S. Elloumi, Using a mixed integer quadratic programming solver for the unconstrained quadratic 0-1 problem, Math. Program., 109 (2007), pp. 55–68.Google Scholar
- 3.R. Freling, R. Lentink, L. Kroon, and D. Huisman, Shunting of passenger train units in a railway station. ERIM Report Series Research in Management, 2002.Google Scholar
- 4.G. Gallo and F. Di Miele, Dispatching buses in parking depots, Transportation Science, 35 (2001), pp. 322–330.Google Scholar
- 5.M. Hamdouni, F. Soumis, and G. Desaulniers, Dispatching buses in a depot minimizing mismatches. 7th IMACS, Scientific Computing Toronto, Canada, 2005.Google Scholar
- 6.P. Hammer and A. Rubin, Some remarks on quadratic programming with 0-1 variables, Revue Francaise d’Informatique et de Recherche Operationelle, 4 (1970), pp. 67–79.Google Scholar
- 7.R. S. Hansmann and U. T. Zimmermann, Optimal Sorting of Rolling Stock at Hump Yards, in Mathematics – Key Technology for the Future: Joint Projects Between Universities and Industry, Springer, Berlin, 2008, pp. 189–203.Google Scholar
- 8.ILOG, CPLEX website. http://www.ilog.com/products/cplex/.
- 9.L. Kaufmann and F. Broeckx, An algorithm for the quadratic assignment problem, European J. Oper. Res., 2 (1978), pp. 204–211.Google Scholar
- 10.L. Kroon, R. Lentink, and A. Schrijver, Shunting of passenger train units: an integrated approach, ERIM Report Series Reference No. ERS-2006-068-LIS, (2006).Google Scholar
- 11.I. Nowak, Relaxation and Decomposition Methods for Mixed Integer Nonlinear Programming, Birkhäuser Verlag, 2005.Google Scholar
- 12.S. Vigerske, Nonconvex mixed-integer nonlinear programming. http://www.math.hu-berlin.de/~stefan/B19/.
- 13.T. Winter, Online and Real-Time Dispatching Problems, PhD thesis, TU Braunschweig, 1998.Google Scholar
- 14.T. Winter and U. Zimmermann, Real-time dispatch of trams in storage yards, Annals of Operations Research, 96 (2000), pp. 287–315.Google Scholar