Abstract
In this paper, we introduce a GRASP for the solution of general linear integer problems. The strategy is based on the separation of the set of variables into the integer subset and the continuous subset. The integer variables are fixed by GRASP and replaced in the original linear problem. If the original problem had continuous variables, it becomes a pure continuous problem, which can be solved by a linear program solver to determine the objective value corresponding to the fixed variables. If the original problem was a pure integer problem, simple algebraic manipulations can be used to determine the objective value that corresponds to the fixed variables. When we assign values to integer variables that lead to an impossible linear problem, the evaluation of the corresponding solution is given by the sum of infeasibilities, together with an infeasibility flag. We report results obtained for some standard benchmark problems, and compare them to those obtained by branch-and-bound and to those obtained by an evolutionary solver.
Keywords
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
Michel R. Berkelaar and Jeroen Dirks. 1p_solve - a solver for linear programming problems with a callable subroutine library. Internet repository, version 2.2, 1994. ftp://ftp.es.ele.tue.nl/pub/lp solve: Last visited on March 25, 2003.
S. Binato, W. J. Henry, D. Loewenstern, and M. G. C. Resende. A greedy randomized adaptive search procedure for job scheduling. In C. C. Ribeiro and P. Hansen, editors, Essays and surveys on metaheuristics, pages 58–79. Kluwer Academic Publishers, 2002.
Robert E. Bixby, Sebastiàn Ceria, Cassandra M. McZeal, and Martin W. P. Savelsbergh. An updated mixed integer programming library. Technical report, Rice University, 1998. TR98–03.
C. Carreto and B. Baker. A GRASP interactive approach to the vehicle routing problem with backhauls. In C. C. Ribeiro and P. Hansen, editors, Essays and Surveys on Metaheuristics, pages 185–199. Kluwer Academic Publishers, 2002.
T. A. Feo and M. G. C. Resende. A probabilistic heuristic for a computacionally difficult set covering problem. Operations Research Letters, 8: 67–71, 1989.
T. A. Feo and M. G. C. Resende. Greedy randomized adaptive search procedures. J. of Global Optimization, 6: 109–133, 1995.
T. A. Feo, K. Venkatraman, and J. F. Bard. A GRASP for a difficult single machine sheduling problem. Computers Operations Research, 18: 635–643, 1991.
P. Festa and M. G. C. Resende. GRASP: an annotated bibliography. In C. C. Ribeiro and P. Hansen, editors, Essays and Surveys on Metaheuristics, pages 325–367. Kluwer Academic Publishers, 2002.
Pierre Hansen and Nenad Mladenovic. Variable neighborhood search: Principles and applications. European Journal of Operational Research, 130: 449–467, 2001.
G. Kontoravdis and J. F. Bard. A GRASP for the vehicle routing problem with time windows. ORSA J. on Computing, 7: 10–23, 1995.
Thomas Lengauer. Combinatorial Algorithms for Integrated Circuit Layout, chapter 8, pages 427–446. Applicable Theory in Computer Science. John Wiley and Sons, 1990.
Y. Li, P. M. Pardalos, and M. G. C. Resende. A greedy randomized adaptive search procedure for the quadratic assignment problem. In P.M. Pardalos and H. Wolkowicz, editors, Quadratic assignment and related problems, volume 16 of DIMACS Series on Discrete Mathematics and Theoretical Computer Science, pages 237–261. American Mathematical Society, 1994.
George L. Nemhauser and Laurence A. Wolsey. Integer and Combinatorial Optimization. Wiley-Interscience in Discrete Mathematics and Optimization, 1988.
Joao P. Pedroso. An evolutionary solver for linear integer programming. BSIS Technical Report 98–7, Riken Brain Science Institute, Wako-shi, Saitama, Japan, 1998.
Joao P. Pedroso. An evolutionary solver for pure integer linear programming. International Transactions in Operational Research, 9 (3): 337–352, May 2002.
L. S. Pitsoulis and M. G. C. Resende. Greedy randomized adaptive search procedures. In P. M. Pardalos and M. G. C. Resende, editors, Handbook of Applied Optimization, pages 168–183. Oxford University Press, 2002.
M. G. C. Resende and T. A. Feo. A GRASP for satisfiability. In D. S. Johnson and M. A. Trick, editors, Cliques, Coloring and Satisfiability: The second DIMACS Implementation Challenge, volume 26 of DIMACS Series on Discrete Mathematics and Theoretical Computer Science, pages 499–520. American Mathematical Society, 1996.
M. G. C. Resende, L. S. Pitsoulis, and P. M. Pardalos. Approximate solution of weighted MAX-SAT problems using GRASP. In Satisfiability problems, volume 35 of DIMACS Series on Discrete Mathematics and Theoretical Computer Science, pages 393–405. American Mathematical Society, 1997.
M. G. C. Resende and C. C. Ribeiro. Greedy randomized adaptive search procedure. In F. Glover and G. Kochenberger, editors, Handbook of Metaheuristics, pages 219–249. Kluwer Academic Publishers, 2002.
A.J. Robertson. A set of greedy randomized adaptive local search procedure (GRASP) implementations for the multidimensional assignment problem. Computational Optimization and Applications, 19: 145–164, 2001.
Laurence A. Wolsey. Integer Programming. Wiley-Interscience in Discrete Mathematics and Optimization, 1998.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media New York
About this chapter
Cite this chapter
Neto, T., Pedroso, J.P. (2003). GRASP for Linear Integer Programming. In: Metaheuristics: Computer Decision-Making. Applied Optimization, vol 86. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-4137-7_26
Download citation
DOI: https://doi.org/10.1007/978-1-4757-4137-7_26
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-5403-9
Online ISBN: 978-1-4757-4137-7
eBook Packages: Springer Book Archive