Advertisement

Conditional Markov Chain Search for the Simple Plant Location Problem Improves Upper Bounds on Twelve Körkel–Ghosh Instances

  • Daniel Karapetyan
  • Boris Goldengorin
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 139)

Abstract

We address a family of hard benchmark instances for the Simple Plant Location Problem (also known as the Uncapacitated Facility Location Problem). The recent attempt by Fischetti et al. Manag Sci 63(7): 2146–2162 (2017) to tackle the Körkel–Ghosh instances resulted in seven new optimal solutions and 22 improved upper bounds. We use automated generation of heuristics to obtain a new algorithm for the Simple Plant Location Problem. In our experiments, our new algorithm matched all the previous best known and optimal solutions, and further improved 12 upper bounds, all within shorter time budgets compared to the previous efforts.

Our algorithm design process is split into two phases: (1) development of algorithmic components such as local search procedures and mutation operators, and (2) composition of a metaheuristic from the available components. Phase (1) requires human expertise and often can be completed by implementing several simple domain-specific routines known from the literature. Phase (2) is entirely automated by employing the Conditional Markov Chain Search (CMCS) framework. In CMCS, a metaheuristic is flexibly defined by a set of parameters, called configuration. Then the process of composition of a metaheuristic from the algorithmic components is reduced to an optimisation problem seeking the best performing CMCS configuration.

We discuss the problem of comparing configurations, and propose a new efficient technique to select the best performing configuration from a large set. To employ this method, we restrict the original CMCS to a simple deterministic case that leaves us with a finite and manageable number of meaningful configurations.

References

  1. 1.
    M.L. Alves, M.T. Almeida, Simulated annealing algorithm for the simple plant location problem: a computational study. Revista Invest. Oper. 12, (1992)Google Scholar
  2. 2.
    E. Balas, M.W. Padberg, On the set covering problem. Oper. Res. 20, 1152–1161 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    M. Balinski, Integer programming: methods, uses, computations. Manag. Sci. 12, 253–313 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    F.F. Barahona, F.N.A. Chudak, Solving large scale uncapacitated facility-location problems, in Approximation and Complexity in Numerical Optimization, ed. by P.M. Pardalos (Kluwer Academic Publishers, Norwell, MA, 1990), pp. 48–62Google Scholar
  5. 5.
    F.F. Barahona, F.N.A. Chudak, Near-optimal solutions to large-scale facility location problems. Discret. Optim. 2, 35–50 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    J.E. Beasley, Lagrangian heuristics for location problems. Eur. J. Oper. Res. 65, 383–399 (1993)CrossRefzbMATHGoogle Scholar
  7. 7.
    C. Beltran-Royo, J.-P. Vial, A. Alonso-Ayuso, Semi-Lagrangian relaxation applied to the uncapacitated facility location problem. Comput. Optim. Appl. 51, 387–409 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    L. Cánovas, M. Landete, A. Marin, On the facets of the simple plant location packing polytope. Discret. Appl. Math. 23, 27–53 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    D.C. Cho, E.J. Johnson, M.W. Padberg, M.R. Rao, On the uncapacitated location problem I: valid inequalities and facets. Math. Oper. Res. 8, 579–589 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    D.C. Cho, E.J. Johnson, M.W. Padberg, M.R. Rao, On the uncapacitated location problem II: facets and lifting theorems. Math. Oper. Res. 8, 590–612 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    G. Cornuejols, G. Nemhauser, L.A. Wolsey, The uncapacitated facility location problem, in Discrete Location Theory, ed. by P.B. Mirchandani, R.L. Francis (Wiley-Interscience, New York, 1990)zbMATHGoogle Scholar
  12. 12.
    G. Cornuejols, J.-M. Thizy, A primal approach to the simple plant location problem. SIAM J. Algebraic Discret. Methods 3(4), 504–510 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    M.S. Daskin, Network and Discrete Location: Models, Algorithms, and Applications, 2nd edn. (Wiley, New York, 2013)zbMATHGoogle Scholar
  14. 14.
    I.R. de Farias, A family of facets for the uncapacitated p-median polytope. Oper. Res. Lett. 28, 161–167 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    D. Erlenkotter, A dual-based procedure for uncapacitated facility location. Oper. Res. 26, 992–1009 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    M. Fischetti, I. Ljubić, M. Sinnl, Redesigning benders decomposition for large-scale facility location. Manag. Sci. 63(7), 2146–2162 (2017)CrossRefGoogle Scholar
  17. 17.
    L. Galli, A.N. Letchford, S.J. Miller, New valid inequalities and facets for the simple plant location problem. Eur. J. Oper. Res. 269(3), 824–833 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    D. Ghosh, Neighborhood search heuristics for the uncapacitated facility location problem. Eur. J. Oper. Res. 150(4), 150–162 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    B. Goldengorin, Data correcting approach for routing and location in networks, in Handbook of Combinatorial Optimization, ed. by P.M. Pardalos, D.-Z. Du, R.L. Graham (Springer, New York, 2013), pp. 929–993CrossRefGoogle Scholar
  20. 20.
    B. Goldengorin, D. Krushinsky, P.M. Pardalos, Cell Formation in Industrial Engineering: Theory, Algorithms and Experiments (Springer, New York, 2013)CrossRefzbMATHGoogle Scholar
  21. 21.
    B. Goldengorin, G.A. Tijssen, D. Ghosh, G. Sierksma, Solving the simple plant location problems using a data correcting approach. J. Glob. Optim. 25, 377–406 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    S. Guha, S. Khuller, Greedy strikes back: improved facility location algorithms. J. Algorithms 31, 228–248 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    M. Guignard, A Lagrangean dual ascent algorithm for simple plant location problems. Eur. J. Oper. Res. 33, 193–200 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    P. Hansen, N. Mladenović, Variable neighborhood search for the p-median. Locat. Sci. 5(4), 207–226 (1997)CrossRefzbMATHGoogle Scholar
  25. 25.
    M. Hoefer, Experimental comparison of heuristic and approximation algorithms for uncapacitated facility location, in Experimental and Efficient Algorithms (Springer, Berlin, 2003), pp. 165–178zbMATHGoogle Scholar
  26. 26.
    K. Jain, M. Mahdian, E. Markakis, A. Saberi, V.V. Vazirani. Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP. J. ACM 60, 795–824 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    D. Karapetyan, A.P. Punnen, A.J. Parkes, Markov chain methods for the bipartite boolean quadratic programming problem. Eur. J. Oper. Res. 260(2), 494–506 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    M. Körkel, On the exact solution of large-scale simple plant location problems. Eur. J. Oper. Res. 39, 157–173 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    J. Kratica, D. Tosic, V. Filipović, I. Ljubić, Solving the simple plant location problem by genetic algorithm. RAIRO Oper. Res. 35, 127–142 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    A.A. Kuehn, M.J. Hamburger, A heuristic program for locating warehouses. Manag. Sci. 9(4), 643–646 (1963)CrossRefGoogle Scholar
  31. 31.
    A. Letchford, S. Miller, An aggressive reduction scheme for the simple plant location problem. Eur. J. Oper. Res. 234, 674–682 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    L. Michel, P. Van Hentenryck, Solving the simple plant location problem by genetic algorithm. RAIRO Oper. Res. 35, 127–142 (2001)MathSciNetCrossRefGoogle Scholar
  33. 33.
    J.G. Morris, On the extent to which certain fixed charge depot location problems can be solved by LP. J. Oper. Res. Soc. 29, 71–76 (1978)CrossRefGoogle Scholar
  34. 34.
    M. Posta, J.A. Ferland, P. Michelon, An exact cooperative method for the uncapacitated facility location problem. Math. Programm. Comput. 6, 199–231 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    M.G.C. Resende, R. Werneck, A hybrid multistart heuristic for the uncapacitated facility location problem. Eur. J. Oper. Res. 174, 54–68 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    L. Schrage, Implicit representation of variable upper bounds in linear programming. Math. Programm. Study 4, 118–132 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    M. Sun, Solving uncapacitated facility location problems using tabu search. Comput. Oper. Res. 33, 2563–2589 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    V.A. Trubin, On a method of solution of integer programming problems of a special kind. Soviet Math. Doklady 10, 1544–1546 (1969)zbMATHGoogle Scholar
  39. 39.
    A. Weber, Theory of the Location of Industries. The University of Chicago Press Chicago, Illinois (1929). English Edition, with Introduction and Notes by Carl J. Friedrich.Google Scholar
  40. 40.
    V. Yigit, M.E. Aydin, O. Turkbey, Evolutionary simulated annealing algorithms for uncapacitated facility location problems, in Adaptive Computing in Design and Manufacture VI, ed. by I.C. Parmee (Springer, New York, 2004), pp. 185–194CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Daniel Karapetyan
    • 1
  • Boris Goldengorin
    • 2
  1. 1.Institute for Analytics and Data ScienceUniversity of EssexColchesterUK
  2. 2.Department of Information Systems and Decision Science, Merrick School of BusinessUniversity of BaltimoreBaltimoreUSA

Personalised recommendations