Matheuristics pp 135-158 | Cite as

Decomposition Techniques as Metaheuristic Frameworks

  • Marco Boschetti
  • Vittorio Maniezzo
  • Matteo Roffilli
Part of the Annals of Information Systems book series (AOIS, volume 10)


Decomposition techniques are well-known as a means for obtaining tight lower bounds for combinatorial optimization problems, and thus as a component for solution methods. Moreover a long-established research literature uses them for defining problem-specific heuristics. More recently it has been observed that they can be the basis also for designing metaheuristics. This tutorial elaborates this last point, showing how the three main decomposition techniques, namely Dantzig-Wolfe, Lagrangean and Benders decompositions, can be turned into model-based, dual-aware metaheuristics. A well known combinatorial optimization problem, the Single Source Capacitated Facility Location Problem, is then chosen for validation, and the implemented codes of the proposed algorithms are benchmarked on standard instances from literature.


Master Problem Lagrangean Relaxation Decomposition Technique Facility Location Problem Bender Decomposition 
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.


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  1. 1.
    M. Agar and S. Salhi. Lagrangean heuristics applied to a variety of large capacitated plant location problems. Journal of the Operational Research Society, 49:1072–1084, 1998.Google Scholar
  2. 2.
    R.K. Ahuja, J.B. Orlin, S. Pallottino, M.P. Scaparra, and M.G. Scutellà. A multi-exchange heuristic for the single source capacitated facility location problem. Management Science, (6):749–760, 2003.Google Scholar
  3. 3.
    A. Atamtürk, G. Nemhauser, and M.W.P. Savelsbergh. A combined Lagrangian, linear programming, and implication heuristic for large-scale set partitioning problems. Journal of Heuristics, 1:247–259, 1996.CrossRefGoogle Scholar
  4. 4.
    F. Barahona and R. Anbil. The volume algorithm: producing primal solutions with a subgradient method. Mathematical Programming, 87:385–399, 2000.CrossRefGoogle Scholar
  5. 5.
    J. Barcelo and J. Casanovas. A heuristic Lagrangean algorithm for the capacitated plant location problem. European Journal of Operational Research, 15:212–226, 1984.CrossRefGoogle Scholar
  6. 6.
    M.S. Bazaraa, J. Jarvis, and H.D. Sherali. Linear Programming and Network Flows. John Wiley & Sons, 1990.Google Scholar
  7. 7.
    J.E. Beasley. Lagrangean relaxation. In C.R. Reeves, editor, Modern heuristic techniques for combinatorial problems, pages 243–303. Blackwell Scientific Publ., 1993.Google Scholar
  8. 8.
    J.F. Benders. Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik, 4:280–322, 1962.CrossRefGoogle Scholar
  9. 9.
    M.A. Boschetti and V. Maniezzo. Benders decomposition, Lagrangean relaxation and metaheuristic design. Journal of Heuristics, 15(3):283–312, 2009.CrossRefGoogle Scholar
  10. 10.
    M.A. Boschetti, V. Maniezzo, and M. Roffilli. A fully distributed Lagrangean solution for a P2P overlay network design problem. Submitted for publication, 2009.Google Scholar
  11. 11.
    M.A. Boschetti, A. Mingozzi, and S. Ricciardelli. An exact algorithm for the simplified multi depot crew scheduling problem. Annals of Operations Research, 127:177–201, 2004.CrossRefGoogle Scholar
  12. 12.
    M.A. Boschetti, A. Mingozzi, and S. Ricciardelli. A dual ascent procedure for the set partitioning problem. Discrete Optimization, 5(4):735–747, 2008.CrossRefGoogle Scholar
  13. 13.
    A. Caprara, M. Fischetti, and P. Toth. A heuristic method for the set covering problem. Operations Research, 47:730–743, 1999.CrossRefGoogle Scholar
  14. 14.
    S. Ceria, P. Nobili, and A. Sassano. A Lagrangian-based heuristic for large-scale set covering problems. Mathematical Programming, 81:215–228, 1995.Google Scholar
  15. 15.
    F.A. Chudak and D.B. Shmoys. Improved approximation algorithms for a capacitated facility location problem. In Proc. 10th Annu. ACM-SIAM Sympos. Discrete Algorithms, pages S875–S876, 1999.Google Scholar
  16. 16.
    G.B. Dantzig and P. Wolfe. Decomposition principle for linear programs. Operations Research, 8:101–111, 1960.CrossRefGoogle Scholar
  17. 17.
    H. Delmaire, J.A. Diaz, E. Fernandez, and M. Ortega. Reactive GRASP and tabu search based heuristics for the single source capacitated plant location problem. INFOR, 37:194–225, 1999.Google Scholar
  18. 18.
    R. Freling, D. Huisman, and A.P.M. Wagelmans. Models and algorithms for integration of vehicle and crew scheduling. Journal of Scheduling, 6:63–85, 2003.CrossRefGoogle Scholar
  19. 19.
    K.L. Hoffman and M. Padberg. Solving airline crew scheduling problems by branch-and-cut. Management Science, 39:657–682, 1993.CrossRefGoogle Scholar
  20. 20.
    K. Holmberg, M. Ronnqvist, and D. Yuan. An exact algorithm for the capacitated facility location problems with single sourcing. European Journal of Operational Research, 113:544–559, 1999.CrossRefGoogle Scholar
  21. 21.
    J. Klincewicz and H. Luss. A Lagrangean relaxation heuristic for capacitated facility location with single-source constraints. Journal of the Operational Research Society, 37:495–500, 1986.Google Scholar
  22. 22.
    V. Maniezzo, M.A. Boschetti, and M. Jelasity. A fully distributed Lagrangean metaheuristic for a P2P overlay network design problem. In Proceedings of the 6th Metaheuristics International Conference (MIC 2005), Vienna, Austria, 2005.Google Scholar
  23. 23.
    S. Martello and P. Toth. Knapsack Problems: Algorithms and Computer implementations. John Wiley, 1990.Google Scholar
  24. 24.
    A. Mingozzi, M.A. Boschetti, S. Ricciardelli, and L. Bianco. A set partitioning approach to the crew scheduling problem. Operations Research, 47:873–888, 1999.CrossRefGoogle Scholar
  25. 25.
    A. Neebe and M. Rao. An algorithm for the fixed-charge assigning users to sources problem. Journal of the Operational Research Society, 34:1107–1113, 1983.Google Scholar
  26. 26.
    H. Pirkul. Efficient algorithm for the capacitated concentrator location problem. Computers & Operations Research, 14:197–208, 1987.CrossRefGoogle Scholar
  27. 27.
    B.T. Polyak. Minimization of unsmooth functionals. USSR Computational Mathematics and Mathematical Physics, 9:14–29, 1969.CrossRefGoogle Scholar
  28. 28.
    M. Ronnqvist, S. Tragantalerngsak, and J. Holt. A repeated matching heuristic for the single source capacitated facility location problem. European Journal of Operational Research, 116:51–68, 1999.CrossRefGoogle Scholar
  29. 29.
    H.D. Sherali and G. Choi. Recovery of primal solutions when using subgradient optimization methods to solve Lagrangian duals of linear programs. Operations Research Letters, 19:105–113,1996.CrossRefGoogle Scholar
  30. 30.
    M. Solomon. Algorithms for the vehicle routing and scheduling problem with time window constraints. Operations Research, 35:254–365, 1987.CrossRefGoogle Scholar
  31. 31.
    R. Sridharan. A Lagrangian heuristic for the capacitated plant location problem with single source constraints. European Journal of Operational Research, 66:305–312, 1991.CrossRefGoogle Scholar
  32. 32.
    M.G.C. Van Krieken, H. Fleuren, and R. Peeters. A Lagrangean relaxation based algorithm for solving set partitioning problems. Technical Report 2004-44, CentER Discussion Paper, 2004.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Marco Boschetti
    • 1
  • Vittorio Maniezzo
    • 2
  • Matteo Roffilli
    • 2
  1. 1.Department of MathematicsUniversity of BolognaBolognaItaly
  2. 2.Department of Computer ScienceUniversity of BolognaBolognaItaly

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