Skip to main content
SpringerLink
Log in

Tree Search and Simulation

  • Chapter
  • First Online:
Applied Simulation and Optimization

Abstract

This chapter presents a general methodology for embodying simulation as part of a tree search procedure, as a technique for solving practical problems in combinatorial optimization. Target problems are either difficult to express as mixed integer optimization models, or have models which provide rather loose bounds; in both cases, traditional, exact methods typically fail. The idea then is to have tree search instantiating part of the variables in a systematic way, and for each particular instantiation—i.e., a node in the search tree—resort to a simulation for assigning values to the remaining variables; then, use the outcome of the simulation for evaluating that node in the tree. This method has been used with considerable success in gameplaying, but has received very limited attention as a tool for optimization. Nevertheless, it has great potential, either as a way for improving known heuristics or as an alternative to metaheuristics. We depart from repeated, randomized simulation based on problem-specific heuristics for applications in scheduling, logistics, and packing, and show how the systematic search in a tree improves the results that can be obtained.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Land, A.H., Doig, A.G.: An automatic method of solving discrete programming problems. Econometrica 28 (1960) 497–520

    Google Scholar 

  2. Lawler, E.L., Wood, D.E.: Branch-and-bound methods: A survey. Operations Research 14 (1966) 699–719

    Google Scholar 

  3. Buchheim, C., Caprara, A., Lodi, A.: An effective branch-and-bound algorithm for convex quadratic integer programming. Mathematical Programming 135 (2012) 369–395

    Google Scholar 

  4. Ng, C., Wang, J.B., Cheng, T.E., Liu, L.: A branch-and-bound algorithm for solving a two-machine flow shop problem with deteriorating jobs. Computers & Operations Research 37 (2010) 83–90

    Google Scholar 

  5. Bazin, J., Li, H., Kweon, I.S., Demonceaux, C., Vasseur, P., Ikeuchi, K.: A branch-and-bound approach to correspondence and grouping problems. IEEE Transactions on Pattern Analysis and Machine Intelligence 35 (2013) 1565–1576

    Google Scholar 

  6. Delling, D., Goldberg, A.V., Razenshteyn, I., Werneck, R.F.: Exact combinatorial branch-and-bound for graph bisection. In: Proceedings of the 14th Meeting on Algorithm Engineering and Experiments (ALENEX’12), Society for Industrial and Applied Mathematics (2012) 30–44

    Google Scholar 

  7. Ginsberg, M.L., Harvey, W.D.: Iterative broadening. Artificial Intelligence 55 (1992) 367–383

    Google Scholar 

  8. Hart, P.E., Nilsson, N.J., Raphael, B.: A formal basis for the heuristic determination of minimum cost paths. IEEE Transactions on Systems Science and Cybernetics 4 (1968) 100–107

    Google Scholar 

  9. Korf, R.: Depth-first iterative-deepening: An optimal admissible tree search. Artificial Intelligence 27 (1985) 97–109

    Google Scholar 

  10. Gomes, C.P., Selman, B., Crato, N., Kautz, H.: Heavy-tailed phenomena in satisfiability and constraint satisfaction problems. Journal of Automated Reasoning 24 (2000) 67–100

    Google Scholar 

  11. Fischetti, M., Monaci, M.: Exploiting erraticism in search. Operations Research 62 (2014) 114–122

    Google Scholar 

  12. Bouzy, B.: Associating shallow and selective global tree search with Monte Carlo for 9\(\times \) 9 Go. In: Computers and Games. Springer (2006) 67–80

    Google Scholar 

  13. Juille, H.R.: Methods for Statistical Inference: Extending the Evolutionary Computation Paradigm. PhD thesis, Waltham, MA, USA (1999)

    Google Scholar 

  14. Coulom, R.: Efficient selectivity and backup operators in Monte-Carlo tree search. In: Proceedings of the 5th International Conference on Computers and Games. CG’06, Berlin, Heidelberg, Springer-Verlag (2007) 72–83

    Google Scholar 

  15. Winands, M.H., Björnsson, Y., Saito, J.T.: Monte-Carlo tree search solver. In: Computers and Games. Springer (2008) 25–36

    Google Scholar 

  16. Takeuchi, S., Kaneko, T., Yamaguchi, K.: Evaluation of Monte Carlo tree search and the application to Go. In: 2008 IEEE Symposium On Computational Intelligence and Games (CIG), IEEE (2008) 191–198

    Google Scholar 

  17. Gelly, S., Kocsis, L., Schoenauer, M., Sebag, M., Silver, D., Szepesvári, C., Teytaud, O.: The grand challenge of computer Go: Monte Carlo tree search and extensions. Communications of the ACM 55 (2012) 106–113

    Google Scholar 

  18. Browne, C.B., Powley, E., Whitehouse, D., Lucas, S.M., Cowling, P.I., Rohlfshagen, P., Tavener, S., Perez, D., Samothrakis, S., Colton, S.: A survey of Monte Carlo tree search methods. IEEE Transactions on Computational Intelligence and AI in Games 4 (2012) 1–43

    Google Scholar 

  19. Sabharwal, A., Samulowitz, H., Reddy, C.: Guiding combinatorial optimization with UCT. In: Proceedings of the 9th International Conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems (CPAIOR 2012), Springer (2012) 356–361

    Google Scholar 

  20. Kocsis, L., Szepesvári, C.: Bandit based Monte-Carlo planning. In Fürnkranz, J., Scheffer, T., Spiliopoulou, M., eds.: Machine Learning: ECML 2006. Volume 4212 of Lecture Notes in Computer Science. Springer, Berlin Heidelberg (2006) 282–293

    Google Scholar 

  21. Auer, P., Cesa-Bianchi, N., Fischer, P.: Finite-time analysis of the multiarmed bandit problem. Machine Learning 47 (2002) 235–256

    Google Scholar 

  22. Garey, M.R., Johnson, D.S.: Computers and Intractability. W. H. Freeman, New York (1979)

    Google Scholar 

  23. Mertens, S.: The easiest hard problem: Number partitioning. In Percus, A., Istrate, G., Moore, C., eds.: Computational Complexity and Statistical Physics, New York, Oxford University Press (2006) 125–139

    Google Scholar 

  24. Mertens, S.: Phase transition in the number partitioning problem. Physical Review Letters 81 (1998) 4281–4284

    Google Scholar 

  25. Alidaee, B., Glover, F., Kochenberger, G.A., Rego, C.: A new modeling and solution approach for the number partitioning problem. Journal of Applied Mathematics and Decision Sciences 9 (2005) 113–121

    Google Scholar 

  26. Koyutürk, M., Aykanat, C.: Iterative-improvement-based declustering heuristics for multi-disk databases. Information Systems 30 (2005) 47–70

    Google Scholar 

  27. Karmarkar, N., Karp, R.: The differencing method of set partitioning. Technical Report UCB/CSD 82/113, University of California - Berkeley, Computer Science Division (1982)

    Google Scholar 

  28. Korf, R.E.: A complete anytime algorithm for number partitioning. Artificial Intelligence 106 (1998) 181–203

    Google Scholar 

  29. Pedroso, J.P., Kubo, M.: Heuristics and exact methods for number partitioning. European Journal of Operational Research 202 (2010) 73–81

    Google Scholar 

  30. Dekker, R., Voogd, P., Asperen, E.: Advanced methods for container stacking. In Kim, K.H., Günther, H.O., eds.: Container Terminals and Cargo Systems. Springer, Berlin Heidelberg (2007) 131–154

    Google Scholar 

  31. Hartmann, S.: A general framework for scheduling equipment and manpower at container terminals. OR Spectrum 26 (2004) 51–74

    Google Scholar 

  32. Avriel, M., Penn, M., Shpirer, N.: Container ship stowage problem: Complexity and connection to the coloring of circle graphs. Discrete Applied Mathematics 103 (2000) 271–279

    Google Scholar 

  33. Avriel, M., Penn, M., Shpirer, N., Witteboon, S.: Stowage planning for container ships to reduce the number of shifts. Annals of Operations Research 76 (1998) 55–71

    Google Scholar 

  34. Rei, R.J., Pedroso, J.P.: Tree search for the stacking problem. Annals of Operations Research 203 (2013) 371–388

    Google Scholar 

  35. Kim, K.H., Hong, G.P.: A heuristic rule for relocating blocks. Computers and Operations Research 33 (2006) 940–954

    Google Scholar 

  36. Caserta, M., Voß, S., Sniedovich, M.: Applying the corridor method to a blocks relocation problem. OR Spectrum 33 (2011) 915–929

    Google Scholar 

  37. Pedroso, J.P., Cunha, S., Tavares, J.N.: Recursive circle packing problems. International Transactions in Operational Research (2014). doi:10.1111/itor.12107

  38. Lenstra, J., Rinnooy Kan, A.: Complexity of packing, covering, and partitioning problems. In Schrijver, A., ed.: Packing and Covering in Combinatorics. Mathematisch Centrum, Amsterdam (1979) 275–291

    Google Scholar 

  39. Stephenson, K.: Circle packing: A mathematical tale. Notices of the American Mathematical Society 50 (2003) 1376–1388

    Google Scholar 

  40. Hifi, M., M’Hallah, R.: A literature review on circle and sphere packing problems: Models and methodologies. Advances in Operations Research 2009 (2009) 1–22

    Google Scholar 

  41. Castillo, I., Kampas, F.J., Pintér, J.D.: Solving circle packing problems by global optimization: Numerical results and industrial applications. European Journal of Operational Research 191 (2008) 786–802

    Google Scholar 

Download references

Acknowledgments

This work was partially funded by PhD grant SFRH/BD/66075/2009 from the Portuguese Foundation for Science and Technology (FCT).

Author information

Authors and Affiliations

  1. INESC TEC and DCC-FCUP, Porto, Portugal

    João Pedro Pedroso & Rui Rei

Authors
  1. João Pedro Pedroso
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rui Rei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rui Rei .

Editor information

Editors and Affiliations

  1. Aviation Academy, Amsterdam University of Applied Sciences, Amsterdam, Noord-Holland, The Netherlands

    Miguel Mujica Mota

  2. Facultad de Ingeniería, Universidad Nacional Autónoma de México, Mexico City, Mexico

    Idalia Flores De La Mota

  3. Optimisation Research Group, NICTA, Sydney, New South Wales, Australia

    Daniel Guimarans Serrano

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Pedroso, J.P., Rei, R. (2015). Tree Search and Simulation. In: Mujica Mota, M., De La Mota, I., Guimarans Serrano, D. (eds) Applied Simulation and Optimization. Springer, Cham. https://doi.org/10.1007/978-3-319-15033-8_4

Download citation

Publish with us

Policies and ethics

Search

Navigation