Advertisement

Computational Optimization and Applications

, Volume 69, Issue 2, pp 501–534 | Cite as

A study of the Bienstock–Zuckerberg algorithm: applications in mining and resource constrained project scheduling

  • Gonzalo Muñoz
  • Daniel Espinoza
  • Marcos Goycoolea
  • Eduardo Moreno
  • Maurice Queyranne
  • Orlando Rivera Letelier
Article
  • 236 Downloads

Abstract

We study a Lagrangian decomposition algorithm recently proposed by Dan Bienstock and Mark Zuckerberg for solving the LP relaxation of a class of open pit mine project scheduling problems. In this study we show that the Bienstock–Zuckerberg (BZ) algorithm can be used to solve LP relaxations corresponding to a much broader class of scheduling problems, including the well-known Resource Constrained Project Scheduling Problem (RCPSP), and multi-modal variants of the RCPSP that consider batch processing of jobs. We present a new, intuitive proof of correctness for the BZ algorithm that works by casting the BZ algorithm as a column generation algorithm. This analysis allows us to draw parallels with the well-known Dantzig–Wolfe decomposition (DW) algorithm. We discuss practical computational techniques for speeding up the performance of the BZ and DW algorithms on project scheduling problems. Finally, we present computational experiments independently testing the effectiveness of the BZ and DW algorithms on different sets of publicly available test instances. Our computational experiments confirm that the BZ algorithm significantly outperforms the DW algorithm for the problems considered. Our computational experiments also show that the proposed speed-up techniques can have a significant impact on the solve time. We provide some insights on what might be explaining this significant difference in performance.

Keywords

Column generation Dantzig–Wolfe Optimization RCPSP 

References

  1. 1.
    Alford, C., Brazil, M., Lee, D.: Optimisation in underground mining. In: Weintraub, A., Romero, C., Bjorndal, T., Epstein, R. (eds.) Handbook of Operations Research in Natural Resources, pp. 561–577. Springer, New York (2007)Google Scholar
  2. 2.
    Artigues, C., Demassey, S., Neron, E.: Resource-Constrained Project Scheduling: Models, Algorithms, Extensions and Applications, vol. 37. Wiley, Hoboken (2010)Google Scholar
  3. 3.
    Berthold, T., Heinz, S., Lübbecke, M., Möhring, R., Schulz, J.: A constraint integer programming approach for resource-constrained project scheduling. In: Lodi, A., Milano, M., Toth, P. (eds.) Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, pp. 313–317. Springer, Berlin (2010)Google Scholar
  4. 4.
    Bertsimas, D., Tsitsiklis, J.: Introduction to Linear Optimization, vol. 6. Athena Scientific, Belmont (1997)Google Scholar
  5. 5.
    Bienstock, D., Zuckerberg, M.: A new LP algorithm for precedence constrained production scheduling. Optim. Online. http://www.optimization-online.org/DB_HTML/2009/08/2380.html (2009)
  6. 6.
    Bienstock, D., Zuckerberg, M.: Solving LP relaxations of large-scale precedence constrained problems. In: Proceedings from the 14th Conference on Integer Programming and Combinatorial Optimization (IPCO). Lecture Notes in Computer Science, vol. 6080 pp. 1–14 (2010)Google Scholar
  7. 7.
    Boland, N., Dumitrescu, I., Froyland, G., Gleixner, A.: LP-based disaggregation approaches to solving the open pit mining production scheduling problem with block processing selectivity. Comput. Oper. Res. 36, 1064–1089 (2009)CrossRefMATHGoogle Scholar
  8. 8.
    Brucker, P., Drexl, A., Möhring, R., Neumann, K., Pesch, E.: Resource-constrained project scheduling: notation, classification, models, and methods. Eur. J. Oper. Res. 112(1), 3–41 (1999)CrossRefMATHGoogle Scholar
  9. 9.
    Chicoisne, R., Espinoza, D., Goycoolea, M., Moreno, E., Rubio, E.: A new algorithm for the open-pit mine production scheduling problem. Oper. Res. 60(3), 517–528 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Christofides, N., Alvarez-Valdés, R., Tamarit, J.: Project scheduling with resource constraints: a branch and bound approach. Eur. J. Oper. Res. 29(3), 262–273 (1987)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Dagdelen, K., Johnson, T.: Optimum open pit mine production scheduling by Lagrangian parameterization. In: Proceedings of the 19th International Symposium on the Application of Computers and Operations Research in the Mineral Industry (APCOM) (1986)Google Scholar
  12. 12.
    Dantzig, G., Wolfe, P.: Decomposition principle for linear programs. Oper. Res. 8(1), 101–111 (1960)CrossRefMATHGoogle Scholar
  13. 13.
    Dassault Systèmes: GEOVIA Whittle (2015). http://www.gemcomsoftware.com/products/whittle
  14. 14.
    Debels, D., Vanhoucke, M.: A decomposition-based genetic algorithm for the resource-constrained project-scheduling problem. Oper. Res. 55(3), 457–469 (2007)CrossRefMATHGoogle Scholar
  15. 15.
  16. 16.
    Espinoza, D., Goycoolea, M., Moreno, E., Newman, A.: Minelib: A library of open pit mining problems. Ann. Oper. Res. 206, 93–114 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Fisher, M.: Optimal solution of scheduling problems using Lagrange multipliers: part I. Oper. Res. 21(5), 1114–1127 (1973)CrossRefMATHGoogle Scholar
  18. 18.
    Goycoolea, M., Espinoza, D., Moreno, E., Rivera, O.: Comparing new and traditional methodologies for production scheduling in open pit mining. In: Proceedings of the 37th International Symposium on the Application of Computers and Operations Research in the Mineral Industry (APCOM), pp. 352–359 (2015)Google Scholar
  19. 19.
    Graham, R., Lawler, E., Lenstra, J., Kan, A.: Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann. Discret. Math. 5, 287–326 (1979)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Hartmann, S., Briskorn, D.: A survey of variants and extensions of the resource-constrained project scheduling problem. Eur. J. Oper. Res. 207(1), 1–14 (2010)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Hochbaum, D.: The pseudoflow algorithm: a new algorithm for the maximum-flow problem. Oper. Res. 56, 992–1009 (2008)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Hustrulid, W., Kuchta, K. (eds.): Open Pit Mine Planning and Design. Taylor and Francis, London (2006)Google Scholar
  23. 23.
    Johnson, T.: Optimum open pit mine production scheduling. Ph.D. thesis, Operations Research Department, University of California, Berkeley (1968)Google Scholar
  24. 24.
    Kolisch, R., Sprecher, A.: PSP-Library. Online at http://www.om-db.wi.tum.de/psplib/datamm.html (1997). [Online; accessed March-2015]
  25. 25.
    Kolisch, R., Sprecher, A.: PSPLIB—a project scheduling problem library. Eur. J. Oper. Res. 96(1), 205–216 (1997)CrossRefMATHGoogle Scholar
  26. 26.
    Lambert, W.B., Newman, A.M.: Tailored Lagrangian relaxation for the open pit block sequencing problem. Ann. Oper. Res. 222(1), 419–438 (2014)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Martinez, M., Newman, A.: A solution approach for optimizing long-and short-term production scheduling at LKAB’s Kiruna mine. Eur. J. Oper. Res. 211(1), 184–197 (2011)CrossRefMATHGoogle Scholar
  28. 28.
  29. 29.
  30. 30.
  31. 31.
    Möhring, R., Schulz, A., Stork, F., Uetz, M.: Solving project scheduling problems by minimum cut computations. Manag. Sci. 49(3), 330–350 (2003)CrossRefMATHGoogle Scholar
  32. 32.
    Newman, A., Kuchta, M.: Using aggregation to optimize long-term production planning at an underground mine. Eur. J. Oper. Res. 176(2), 1205–1218 (2007)CrossRefMATHGoogle Scholar
  33. 33.
    Newman, A., Rubio, E., Caro, R., Weintraub, A., Eurek, K.: A review of operations research in mine planning. Interfaces 40, 222–245 (2010)CrossRefGoogle Scholar
  34. 34.
    Osanloo, M., Gholamnejad, J., Karimi, B.: Long-term open pit mine production planning: a review of models and algorithms. Int. J. Min. Reclam. Environ. 22(1), 3–35 (2008)CrossRefGoogle Scholar
  35. 35.
    O’Sullivan, D., Newman, A.: Extraction and backfill scheduling in a complex underground mine. Interfaces 44(2), 204–221 (2014)CrossRefGoogle Scholar
  36. 36.
    O’Sullivan, D., Newman, A., Brickey, A.: Is open pit production scheduling ’easier’ than its underground counterpart? Min. Eng. 67(4), 68–73 (2015)Google Scholar
  37. 37.
    Pessoa, A., Sadyvok, R., Uchoa, E., Vanderbeck, F.: In-out separation and column generation stabilization by dual price smoothing. In: 12th International Symposium on Experimental Algorithms(SEA), Rome, Lecture Notes in Computer Science, vol. 7933, pp. 354–365 (2013)Google Scholar
  38. 38.
    Pritsker, A., Waiters, L., Wolfe, P.: Multiproject scheduling with limited resources: a zero-one programming approach. Manag. Sci. 16(1), 93–108 (1969)CrossRefGoogle Scholar
  39. 39.
    Zhu, G., Bard, J., Yu, G.: A branch-and-cut procedure for the multimode resource-constrained project-scheduling problem. INFORMS J. Comput. 18(3), 377–390 (2006)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Industrial Engineering and Operations ResearchColumbia UniversityNew YorkUSA
  2. 2.Gurobi OptimizationHoustonUSA
  3. 3.School of BusinessUniversidad Adolfo IbañezSantiagoChile
  4. 4.Faculty of EngineeringUniversidad Adolfo IbañezSantiagoChile
  5. 5.School of BusinessUniversity of British ColumbiaVancouverCanada
  6. 6.School of Business and Faculty of EngineeringUniversidad Adolfo IbañezSantiagoChile

Personalised recommendations