A computational study and survey of methods for the single-row facility layout problem

Abstract

The single-row facility layout problem (SRFLP) is an NP-hard combinatorial optimization problem that is concerned with the arrangement of n departments of given lengths on a line so as to minimize the weighted sum of the distances between department pairs. (SRFLP) is the one-dimensional version of the facility layout problem that seeks to arrange rectangular departments so as to minimize the overall interaction cost. This paper compares the different modelling approaches for (SRFLP) and applies a recent SDP approach for general quadratic ordering problems from Hungerländer and Rendl to (SRFLP). In particular, we report optimal solutions for several (SRFLP) instances from the literature with up to 42 departments that remained unsolved so far. Secondly we significantly reduce the best known gaps and running times for large instances with up to 110 departments.

This is a preview of subscription content, log in to check access.

Notes

  1. 1.

    Most of the instances can be downloaded from http://flplib.uwaterloo.ca/.

  2. 2.

    For exact numbers of the speed differences see http://www.cpubenchmark.net/.

  3. 3.

    These instances and the corresponding optimal orderings are available from http://flplib.uwaterloo.ca/.

  4. 4.

    Most of the instances can be downloaded from http://flplib.uwaterloo.ca/. Our improved gaps and the corresponding orderings are also available there.

  5. 5.

    For details see http://www.cpubenchmark.net/.

References

  1. 1.

    Amaral, A.R.S.: On the exact solution of a facility layout problem. Eur. J. Oper. Res. 173(2), 508–518 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Amaral, A.R.S.: An exact approach to the one-dimensional facility layout problem. Oper. Res. 56(4), 1026–1033 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Amaral, A.R.S.: A new lower bound for the single row facility layout problem. Discrete Appl. Math. 157(1), 183–190 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Amaral, A.R.S., Letchford, A.N.: A polyhedral approach to the single row facility layout problem (2011, in preparation). Preprint available from http://www.optimization-online.org/DB_FILE/2008/03/1931.pdf

  5. 5.

    Anjos, M.F., Kennings, A., Vannelli, A.: A semidefinite optimization approach for the single-row layout problem with unequal dimensions. Discrete Optim. 2(2), 113–122 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Anjos, M.F., Lasserre, J. (eds.): Handbook of Semidefinite, Conic and Polynomial Optimization International Series in Operations Research & Management Science. Springer, Berlin (2011, to appear)

  7. 7.

    Anjos, M.F., Liers, F.: Global Approaches for facility layout and vlsi floorplanning. In: Anjos, M.F., Lasserre, J.B. (eds.): Handbook of Semidefinite, Cone and Polynomial Optimization: Theory, Algorithms, Software and Applications (2012, to appear)

  8. 8.

    Anjos, M.F., Vannelli, A.: Computing globally optimal solutions for single-row layout problems using semidefinite programming and cutting planes. INFORMS J. Comput. 20(4), 611–617 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Anjos, M.F., Yen, G.: Provably near-optimal solutions for very large single-row facility layout problems. Optim. Methods Softw. 24(4), 805–817 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Barahona, F., Mahjoub, A.R.: On the cut polytope. Math. Program. 36, 157–173 (1986)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Buchheim, C., Liers, F., Oswald, M.: Speeding up ip-based algorithms for constrained quadratic 0-1 optimization. Math. Program. 124, 513–535 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Buchheim, C., Wiegele, A., Zheng, L.: Exact algorithms for the quadratic linear ordering problem. INFORMS J. Comput. 168–177 (2009)

  13. 13.

    Chimani, M., Hungerländer, P., Jünger, M., Mutzel, P.: An SDP approach to multi-level crossing minimization. In: Proceedings of Algorithm Engineering & Experiments [ALENEX’2011] (2011)

    Google Scholar 

  14. 14.

    Datta, D., Amaral, A.R.S., Figueira, J.R.: Single row facility layout problem using a permutation-based genetic algorithm. Eur. J. Oper. Res. 213(2), 388–394 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Deza, M.M., Laurent, M.: Geometry of Cuts and Metrics. Algorithms and Combinatorics, vol. 15. Springer, Berlin (1997)

    Google Scholar 

  16. 16.

    Fischer, I., Gruber, G., Rendl, F., Sotirov, R.: Computational experience with a bundle method for semidefinite cutten plane relaxations of max-cut and equipartition. Math. Program. 105, 451–469 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete problems. In: STOC’74: Proceedings of the Sixth Annual ACM Symposium on Theory of Computing, New York, pp. 47–63 (1974)

    Google Scholar 

  18. 18.

    Goemans, M., Williamson, D.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Gomes de Alvarenga, A., Negreiros-Gomes, F.J., Mestria, M.: Metaheuristic methods for a class of the facility layout problem. J. Intell. Manuf. 11, 421–430 (2000)

    Article  Google Scholar 

  20. 20.

    Hall, K.M.: An r-dimensional quadratic placement algorithm. Manag. Sci. 17(3), 219–229 (1970)

    MATH  Article  Google Scholar 

  21. 21.

    Hammer, P.: Some network flow problems solved with pseudo-Boolean programming. Oper. Res. 13, 388–399 (1965)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Heragu, S.S., Alfa, A.S.: Experimental analysis of simulated annealing based algorithms for the layout problem. Eur. J. Oper. Res. 57(2), 190–202 (1992)

    MATH  Article  Google Scholar 

  23. 23.

    Heragu, S.S., Kusiak, A.: Machine layout problem in flexible manufacturing systems. Oper. Res. 36(2), 258–268 (1988)

    Article  Google Scholar 

  24. 24.

    Heragu, S.S., Kusiak, A.: Efficient models for the facility layout problem. Eur. J. Oper. Res. 53(1), 1–13 (1991)

    MATH  Article  Google Scholar 

  25. 25.

    Hiriart-Urruty, J.-B., Lemarechal, C.: Convex Analysis and Minimization Algorithms (vols. 1 and 2). Springer, Berlin (1993)

    Google Scholar 

  26. 26.

    Hungerländer, P.: Exact approaches to ordering problems. Ph.D. thesis, Alpen-Adria Universität Klagenfurt (2011)

  27. 27.

    Hungerländer, P., Rendl, F.: Semidefinite relaxations of ordering problems. Math. Program., Ser. B (2011, accepted). Preprint available at http://www.optimization-online.org/DB_HTML/2010/08/2696.html

  28. 28.

    Karp, R.M., Held, M.: Finite-state processes and dynamic programming. SIAM J. Appl. Math. 15(3), 693–718 (1967)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Kumar, K.R., Hadjinicola, G.C., li Lin, T.: A heuristic procedure for the single-row facility layout problem. Eur. J. Oper. Res. 87(1), 65–73 (1995)

    MATH  Article  Google Scholar 

  30. 30.

    Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optim. 1, 166–190 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Love, R.F., Wong, J.Y.: On solving a one-dimensional space allocation problem with integer programming. INFOR, Inf. Syst. Oper. Res. 14, 139–143 (1967)

    Google Scholar 

  32. 32.

    Picard, J.-C., Queyranne, M.: On the one-dimensional space allocation problem. Oper. Res. 29(2), 371–391 (1981)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Rendl, F., Rinaldi, G., Wiegele, A.: Solving max-cut to optimality by intersecting semidefinite and polyhedral relaxations. Math. Program. 212, 307–335 (2010)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Romero, D., Sánchez-Flores, A.: Methods for the one-dimensional space allocation problem. Comput. Oper. Res. 17(5), 465–473 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Samarghandi, H., Eshghi, K.: An efficient tabu algorithm for the single row facility layout problem. Eur. J. Oper. Res. 205(1), 98–105 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Sanjeevi, S., Kianfar, K.: Note: a polyhedral study of triplet formulation for single row facility layout problem. Discrete Appl. Math. 158, 1861–1867 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Simmons, D.M.: One-Dimensional space allocation: an ordering algorithm. Oper. Res. 17, 812–826 (1969)

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Simmons, D.M.: A further note on one-dimensional space allocation. Oper. Res. 19, 249 (1971)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Simone, C.D.: The cut polytope and the Boolean quadric polytope. Discrete Math. 79(1), 71–75 (1990)

    MATH  Article  Google Scholar 

  40. 40.

    Suryanarayanan, J., Golden, B., Wang, Q.: A new heuristic for the linear placement problem. Comput. Oper. Res. 18(3), 255–262 (1991)

    MATH  Article  Google Scholar 

  41. 41.

    Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.): Handbook of Semidefinite Programming. Kluwer Academic, Boston (2000)

    Google Scholar 

Download references

Acknowledgements

We thank three anonymous referees for their constructive comments and suggestions for improvement leading to the present version of the paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Philipp Hungerländer.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hungerländer, P., Rendl, F. A computational study and survey of methods for the single-row facility layout problem. Comput Optim Appl 55, 1–20 (2013). https://doi.org/10.1007/s10589-012-9505-8

Download citation

Keywords

  • Single-row facility layout
  • Space allocation
  • Semidefinite optimization
  • Combinatorial optimization