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Solving Multiobjective Discrete Optimization Problems with Propositional Minimal Model Generation

  • Takehide Soh
  • Mutsunori Banbara
  • Naoyuki Tamura
  • Daniel Le Berre
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10416)

Abstract

We propose a propositional logic based approach to solve MultiObjective Discrete Optimization Problems (MODOPs). In our approach, there exists a one-to-one correspondence between a Pareto front point of MODOP and a \(P\)-minimal model of the CNF formula obtained from MODOP. This correspondence is achieved by adopting the order encoding as CNF encoding for multiobjective functions. Finding the Pareto front is done by enumerating all P-minimal models. The beauty of the approach is that each Pareto front point is blocked by a single clause that contains at most one literal for each objective function. We evaluate the effectiveness of our approach by empirically contrasting it to a state-of-the-art MODOP solving technique.

References

  1. 1.
    Alarcon-Rodriguez, A., Ault, G., Galloway, S.: Multi-objective planning of distributed energy resources: a review of the state-of-the-art. Renew. Sustain. Energy Rev. 14(5), 1353–1366 (2010)CrossRefGoogle Scholar
  2. 2.
    Ansótegui, C., Manyà, F.: Mapping problems with finite-domain variables to problems with boolean variables. In: Hoos, H.H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 1–15. Springer, Heidelberg (2005). doi: 10.1007/11527695_1 CrossRefGoogle Scholar
  3. 3.
    Bailleux, O., Boufkhad, Y.: Efficient CNF encoding of boolean cardinality constraints. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 108–122. Springer, Heidelberg (2003). doi: 10.1007/978-3-540-45193-8_8 CrossRefGoogle Scholar
  4. 4.
    Ballestero, E., Bravo, M., Pérez-Gladish, B., Parra, M.A., Plà-Santamaria, D.: Socially responsible investment: a multicriteria approach to portfolio selection combining ethical and financial objectives. Eur. J. Oper. Res. 216(2), 487–494 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Banbara, M., Matsunaka, H., Tamura, N., Inoue, K.: Generating combinatorial test cases by efficient SAT encodings suitable for CDCL SAT solvers. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR 2010. LNCS, vol. 6397, pp. 112–126. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-16242-8_9 CrossRefGoogle Scholar
  6. 6.
    Bergman, D., Cire, A.A.: Multiobjective optimization by decision diagrams. In: Rueher, M. (ed.) CP 2016. LNCS, vol. 9892, pp. 86–95. Springer, Cham (2016). doi: 10.1007/978-3-319-44953-1_6 CrossRefGoogle Scholar
  7. 7.
    Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability, Frontiers in Artificial Intelligence and Applications (FAIA), vol. 185. IOS Press, Amsterdam (2009)Google Scholar
  8. 8.
    Boland, N., Charkhgard, H., Savelsbergh, M.W.P.: A new method for optimizing a linear function over the efficient set of a multiobjective integer program. Eur. J. Oper. Res. 260(3), 904–919 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Brewka, G., Delgrande, J.P., Romero, J., Schaub, T.: asprin: customizing answer set preferences without a headache. In: Proceedings of the 29th National Conference on Artificial Intelligence (AAAI 2015), pp. 1467–1474 (2015)Google Scholar
  10. 10.
    Burke, E.K., Li, J., Qu, R.: A pareto-based search methodology for multi-objective nurse scheduling. Ann. Oper. Res. 196(1), 91–109 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Crawford, J.M., Baker, A.B.: Experimental results on the application of satisfiability algorithms to scheduling problems. In: Proceedings of the 12th National Conference on Artificial Intelligence (AAAI 1994), pp. 1092–1097 (1994)Google Scholar
  12. 12.
    Deb, K., Agrawal, S., Pratap, A., Meyarivan, T.: A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: NSGA-II. In: Schoenauer, M., Deb, K., Rudolph, G., Yao, X., Lutton, E., Merelo, J.J., Schwefel, H.-P. (eds.) PPSN 2000. LNCS, vol. 1917, pp. 849–858. Springer, Heidelberg (2000). doi: 10.1007/3-540-45356-3_83 CrossRefGoogle Scholar
  13. 13.
    Deb, K., Agrawal, S., Pratap, A., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)CrossRefGoogle Scholar
  14. 14.
    Ehrgott, M.: Multicriteria Optimization. Springer, Heidelberg (2005). doi: 10.1007/3-540-27659-9 zbMATHGoogle Scholar
  15. 15.
    Figueira, J., Greco, S., Ehrgott, M.: Multiple Criteria Decision Analysis: State of the Art Surveys. International Series in Operations Research & Management Science. Springer, Heidelberg (2005). doi: 10.1007/b100605 CrossRefzbMATHGoogle Scholar
  16. 16.
    Gent, I.P., Nightingale, P.: A new encoding of alldifferent into SAT. In: Proceedings of the 3rd International Workshop on Modelling and Reformulating Constraint Satisfaction Problems (2004)Google Scholar
  17. 17.
    Inoue, K., Soh, T., Ueda, S., Sasaura, Y., Banbara, M., Tamura, N.: A competitive and cooperative approach to propositional satisfiability. Discrete Appl. Math. 154(16), 2291–2306 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Iturriaga, S., Dorronsoro, B., Nesmachnow, S.: Multiobjective evolutionary algorithms for energy and service level scheduling in a federation of distributed datacenters. Int. Trans. Oper. Res. 24(1–2), 199–228 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kirlik, G., Sayin, S.: A new algorithm for generating all nondominated solutions of multiobjective discrete optimization problems. Eur. J. Oper. Res. 232(3), 479–488 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Knuth, D.E.: The Art of Computer Programming, Volume 4, Fascicle 6: Satisfiability. Addison-Wesley Professional, Boston (2015)Google Scholar
  21. 21.
    Koshimura, M., Nabeshima, H., Fujita, H., Hasegawa, R.: Minimal model generation with respect to an atom set. In: Proceedings of the the 7th International Workshop on First-Order Theorem Proving (FTP 2009), pp. 49–59 (2009)Google Scholar
  22. 22.
    Laumanns, M., Thiele, L., Zitzler, E.: An efficient, adaptive parameter variation scheme for metaheuristics based on the epsilon-constraint method. Eur. J. Oper. Res. 169(3), 932–942 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lukasiewycz, M., Glaß, M., Haubelt, C., Teich, J.: Solving multi-objective pseudo-boolean problems. In: Marques-Silva, J., Sakallah, K.A. (eds.) SAT 2007. LNCS, vol. 4501, pp. 56–69. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-72788-0_9 CrossRefGoogle Scholar
  24. 24.
    Marinescu, R.: Exploiting problem decomposition in multi-objective constraint optimization. In: Gent, I.P. (ed.) CP 2009. LNCS, vol. 5732, pp. 592–607. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-04244-7_47 CrossRefGoogle Scholar
  25. 25.
    Marinescu, R.: Best-first vs. depth-first AND/OR search for multi-objective constraint optimization. In: Proceedings of the 22nd IEEE International Conference on Tools with Artificial Intelligence (ICTAI 2010), pp. 439–446 (2010)Google Scholar
  26. 26.
    Metodi, A., Codish, M., Lagoon, V., Stuckey, P.J.: Boolean equi-propagation for optimized SAT encoding. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 621–636. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-23786-7_47 CrossRefGoogle Scholar
  27. 27.
    Nabeshima, H., Soh, T., Inoue, K., Iwanuma, K.: Lemma reusing for SAT based planning and scheduling. In: Proceedings of the International Conference on Automated Planning and Scheduling 2006 (ICAPS 2006), pp. 103–112 (2006)Google Scholar
  28. 28.
    Niemelä, I.: A tableau calculus for minimal model reasoning. In: Miglioli, P., Moscato, U., Mundici, D., Ornaghi, M. (eds.) TABLEAUX 1996. LNCS, vol. 1071, pp. 278–294. Springer, Heidelberg (1996). doi: 10.1007/3-540-61208-4_18 CrossRefGoogle Scholar
  29. 29.
    Ohrimenko, O., Stuckey, P.J., Codish, M.: Propagation via lazy clause generation. Constraints 14(3), 357–391 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Okimoto, T., Joe, Y., Iwasaki, A., Matsui, T., Hirayama, K., Yokoo, M.: Interactive algorithm for multi-objective constraint optimization. In: Milano, M. (ed.) CP 2012. LNCS, pp. 561–576. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-33558-7_41 CrossRefGoogle Scholar
  31. 31.
    Ozlen, M., Burton, B.A., MacRae, C.A.G.: Multi-objective integer programming: an improved recursive algorithm. J. Optim. Theory Appl. 160(2), 470–482 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Papadimitriou, C.H., Yannakakis, M.: On the approximability of trade-offs and optimal access of web sources. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science (FOCS 2000), pp. 86–92 (2000)Google Scholar
  33. 33.
    Rollon, E., Larrosa, J.: Bucket elimination for multiobjective optimization problems. J. Heuristics 12(4–5), 307–328 (2006)CrossRefzbMATHGoogle Scholar
  34. 34.
    Rollon, E., Larrosa, J.: Multi-objective russian doll search. In: Proceedings of the 22nd National Conference on Artificial Intelligence (AAAI 2007), pp. 249–254 (2007)Google Scholar
  35. 35.
    Schwind, N., Okimoto, T., Konieczny, S., Wack, M., Inoue, K.: Utilitarian and egalitarian solutions for multi-objective constraint optimization. In: Proceedings of the 26th IEEE International Conference on Tools with Artificial Intelligence (ICTAI 2014), pp. 170–177. IEEE Computer Society (2014)Google Scholar
  36. 36.
    Soh, T., Banbara, M., Tamura, N.: Proposal and evaluation of hybrid encoding of CSP to SAT integrating order and log encodings. Int. J. Artif. Intell. Tools 26(1), 1–29 (2017)CrossRefGoogle Scholar
  37. 37.
    Soh, T., Inoue, K., Tamura, N., Banbara, M., Nabeshima, H.: A SAT-based method for solving the two-dimensional strip packing problem. Fundam. Inf. 102(3–4), 467–487 (2010)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Tamura, N., Banbara, M., Soh, T.: PBSugar: Compiling pseudo-boolean constraints to SAT with order encoding. In: Proceedings of the 25th IEEE International Conference on Tools with Artificial Intelligence (ICTAI 2013), pp. 1020–1027. IEEE, November 2013Google Scholar
  39. 39.
    Tamura, N., Taga, A., Kitagawa, S., Banbara, M.: Compiling finite linear CSP into SAT. Constraints 14(2), 254–272 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Tanjo, T., Tamura, N., Banbara, M.: Sugar++: a SAT-based Max-CSP/COP solver. In: Proceedings of the 3rd International CSP Solver Competition, pp. 77–82 (2008)Google Scholar
  41. 41.
    Ugarte, W., Boizumault, P., Crémilleux, B., Lepailleur, A., Loudni, S., Plantevit, M., Raïssi, C., Soulet, A.: Skypattern mining: from pattern condensed representations to dynamic constraint satisfaction problems. Artif. Intell. 244, 48–69 (2017). https://doi.org/10.1016/j.artint.2015.04.003 MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Wang, L., Ng, A.H.C., Deb, K. (eds.): Multi-objective Evolutionary Optimisation for Product Design and Manufacturing. Springer, Heidelberg (2011). doi: 10.1007/978-0-85729-652-8 Google Scholar
  43. 43.
    Wilson, N., Razak, A., Marinescu, R.: Computing possibly optimal solutions for multi-objective constraint optimisation with tradeoffs. In: Proceedings of the 24th International Joint Conference on Artificial Intelligence (IJCAI 2015), pp. 815–822 (2015)Google Scholar
  44. 44.
    Yi, D., Goodrich, M.A., Seppi, K.D.: MORRF*: sampling-based multi-objective motion planning. In: Proceedings of the 24th International Joint Conference on Artificial Intelligence (IJCAI 2015), pp. 1733–1741 (2015)Google Scholar
  45. 45.
    Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: improving the strength pareto evolutionary algorithm for multiobjective optimization. In: Proceedings of the 4th International Conference of Evolutionary Methods for Design, Optimisation and Control with Application to Industrial Problems (EUROGEN 2001), pp. 95–100 (2002)Google Scholar
  46. 46.
    Zitzler, E., Künzli, S.: Indicator-based selection in multiobjective search. In: Proceedings of the 8th International Conference on Parallel Problem Solving from Nature (PPSN VIII), pp. 832–842 (2004)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Takehide Soh
    • 1
  • Mutsunori Banbara
    • 1
  • Naoyuki Tamura
    • 1
  • Daniel Le Berre
    • 2
  1. 1.Information Science and Technology CenterKobe UniversityKobeJapan
  2. 2.CRIL-CNRS, Université d’ArtoisLensFrance

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