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An Algebraic Approach for the Search Space of Permutations with Repetition

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Book cover Evolutionary Computation in Combinatorial Optimization (EvoCOP 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12102))

Abstract

We present an algebraic approach for dealing with combinatorial optimization problems based on permutations with repetition. The approach is an extension of an algebraic framework defined for combinatorial search spaces which can be represented by a group (in the algebraic sense). Since permutations with repetition does not have the group structure, in this work we derive some definitions and we devise discrete operators that allow to design algebraic evolutionary algorithms whose search behavior is in line with the algebraic framework. In particular, a discrete Differential Evolution algorithm which directly works on the space of permutations with repetition is defined and analyzed. As a case of study, an implementation of this algorithm is provided for the Job Shop Scheduling Problem. Experiments have been held on commonly adopted benchmark suites, and they show that the proposed approach obtains competitive results compared to the known optimal objective values.

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Notes

  1. 1.

    Even in the Euclidean space \(\mathbb {R}^n\), multiplying a vector by a scalar has a geometric meaning only if we interpret this vector as a proper free vector and not as a point in the space.

  2. 2.

    These JSSP instances can be downloaded from the website http://jobshop.jjvh.nl.

References

  1. Baioletti, M., Milani, A., Santucci, V.: Algebraic crossover operators for permutations. In: 2018 IEEE Congress on Evolutionary Computation (CEC 2018), pp. 1–8 (2018). https://doi.org/10.1109/CEC.2018.8477867

  2. Baioletti, M., Milani, A., Santucci, V.: A new precedence-based ant colony optimization for permutation problems. In: Shi, Y., et al. (eds.) SEAL 2017. LNCS, vol. 10593, pp. 960–971. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-68759-9_79

    Chapter  Google Scholar 

  3. Baioletti, M., Milani, A., Santucci, V.: Automatic algebraic evolutionary algorithms. In: Pelillo, M., Poli, I., Roli, A., Serra, R., Slanzi, D., Villani, M. (eds.) WIVACE 2017. CCIS, vol. 830, pp. 271–283. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78658-2_20

    Chapter  MATH  Google Scholar 

  4. Baioletti, M., Milani, A., Santucci, V.: Learning Bayesian networks with algebraic differential evolution. In: Auger, A., Fonseca, C.M., Lourenço, N., Machado, P., Paquete, L., Whitley, D. (eds.) PPSN 2018. LNCS, vol. 11102, pp. 436–448. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-99259-4_35

    Chapter  Google Scholar 

  5. Baioletti, M., Milani, A., Santucci, V.: MOEA/DEP: an algebraic decomposition-based evolutionary algorithm for the multiobjective permutation flowshop scheduling problem. In: Liefooghe, A., López-Ibáñez, M. (eds.) EvoCOP 2018. LNCS, vol. 10782, pp. 132–145. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-77449-7_9

    Chapter  MATH  Google Scholar 

  6. Baioletti, M., Milani, A., Santucci, V.: Variable neighborhood algebraic differential evolution: an application to the linear ordering problem with cumulative costs. Inf. Sci. 507, 37–52 (2020). https://doi.org/10.1016/j.ins.2019.08.016

    Article  Google Scholar 

  7. Baioletti, M., Milani, A., Santucci, V., Bartoccini, U.: An experimental comparison of algebraic differential evolution using different generating sets. In: Proceedings of the Genetic and Evolutionary Computation Conference Companion, GECCO 2019, pp. 1527–534 (2019). https://doi.org/10.1145/3319619.3326854

  8. Bierwirth, C.: A generalized permutation approach to job shop scheduling with genetic algorithms. Oper. Res. Spektrum 17(2), 87–92 (1995)

    Google Scholar 

  9. Bierwirth, C., Mattfeld, D.C., Kopfer, H.: On permutation representations for scheduling problems. In: Voigt, H.-M., Ebeling, W., Rechenberg, I., Schwefel, H.-P. (eds.) PPSN 1996. LNCS, vol. 1141, pp. 310–318. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-61723-X_995

    Chapter  Google Scholar 

  10. Brest, J., Greiner, S., Boskovic, B., Mernik, M., Zumer, V.: Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark problems. IEEE Trans. Evol. Comput. 10(6), 646–657 (2006)

    Article  Google Scholar 

  11. Çaliş, B., Bulkan, S.: A research survey: review of AI solution strategies of job shop scheduling problem. J. Intell. Manuf. 26(5), 961–973 (2015)

    Article  Google Scholar 

  12. Cheng, R., Gen, M., Tsujimura, Y.: A tutorial survey of job-shop scheduling problems using genetic algorithms, Part I: representation. Comput. Ind. Eng. 30(4), 983–997 (1996)

    Article  Google Scholar 

  13. Cheng, R., Gen, M., Tsujimura, Y.: A tutorial survey of job-shop scheduling problems using genetic algorithms, Part II: hybrid genetic search strategies. Comput. Ind. Eng. 36(2), 343–364 (1999)

    Article  Google Scholar 

  14. González, M.Á., Oddi, A., Rasconi, R.: Multi-objective optimization in a job shop with energy costs through hybrid evolutionary techniques. In: Proceedings of the 27th International Conference on Automated Planning and Scheduling (2017)

    Google Scholar 

  15. van Hoorn, J.J.: The current state of bounds on benchmark instances of the job-shop scheduling problem. J. Sched. 21(1), 127–128 (2018)

    Article  MathSciNet  Google Scholar 

  16. Jovanovski, J., Arsov, N., Stevanoska, E., Siljanoska Simons, M., Velinov, G.: A meta-heuristic approach for RLE compression in a column store table. Soft Comput. 23(12), 4255–4276 (2019)

    Article  Google Scholar 

  17. Kim, J., Hwang, I., Kim, Y.H., Moon, B.R.: Genetic approaches for graph partitioning: a survey. In: Proceedings of the GECCO 2011, pp. 473–480. ACM (2011)

    Google Scholar 

  18. Lang, S.: Algebra, vol. 211. Springer, Heidelberg (2002). https://doi.org/10.1007/978-1-4613-0041-0

    Book  MATH  Google Scholar 

  19. Milani, A., Santucci, V.: Asynchronous differential evolution. In: Proceedings of 2010 IEEE Congress on Evolutionary Computation (CEC 2010), pp. 1–7 (2010). https://doi.org/10.1109/CEC.2010.5586107

  20. Moraglio, A., Kim, Y.H., Yoon, Y., Moon, B.R.: Geometric crossovers for multiway graph partitioning. Evol. Comput. 15(4), 445–474 (2007)

    Article  Google Scholar 

  21. Nowicki, E., Smutnicki, C.: An advanced tabu search algorithm for the job shop problem. J. Sched. 8(2), 145–159 (2005)

    Article  MathSciNet  Google Scholar 

  22. Santucci, V., Baioletti, M., Milani, A.: Algebraic differential evolution algorithm for the permutation flowshop scheduling problem with total flowtime criterion. IEEE Trans. Evol. Comput. 20(5), 682–694 (2016). https://doi.org/10.1109/TEVC.2015.2507785

    Article  MATH  Google Scholar 

  23. Santucci, V., Baioletti, M., Milani, A.: Tackling permutation-based optimization problems with an algebraic particle swarm optimization algorithm. Fundamenta Informaticae 167(1–2), 133–158 (2019). https://doi.org/10.3233/FI-2019-1812

    Article  MathSciNet  MATH  Google Scholar 

  24. Santucci, V., Baioletti, M., Di Bari, G., Milani, A.: A binary algebraic differential evolution for the multidimensional two-way number partitioning problem. In: Liefooghe, A., Paquete, L. (eds.) EvoCOP 2019. LNCS, vol. 11452, pp. 17–32. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-16711-0_2

    Chapter  Google Scholar 

  25. Santucci, V., Ceberio, J.: Using pairwise precedences for solving the linear ordering problem. Appl. Soft Comput. 87 (2020). https://doi.org/10.1016/j.asoc.2019.105998

  26. Santucci, V., Milani, A.: Particle swarm optimization in the EDAs framework. In: Gaspar-Cunha, A., Takahashi, R., Schaefer, G., Costa, L. (eds.) Soft Computing in Industrial Applications, pp. 87–96. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-20505-7_7

  27. Sörensen, K.: Distance measures based on the edit distance for permutation-type representations. J. Heuristics 13(1), 35–47 (2007)

    Article  Google Scholar 

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Acknowledgement

The research described in this work has been partially supported by: “Università per Stranieri di Perugia – Finanziamento per Progetti di Ricerca di Ateneo –PRA 2020”, and by RCB-2015 Project “Algoritmi Randomizzati per l’Ottimizzazione e la Navigazione di Reti Semantiche” and RCB-2015 Project “Algoritmi evolutivi per problemi di ottimizzazione combinatorica” of Department of Mathematics and Computer Science of University of Perugia.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Perugia, Perugia, Italy

    Marco Baioletti & Alfredo Milani

  2. Department of Humanities and Social Sciences, University for Foreigners of Perugia, Perugia, Italy

    Valentino Santucci

Authors
  1. Marco Baioletti
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  2. Alfredo Milani
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  3. Valentino Santucci
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Correspondence to Valentino Santucci .

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Editors and Affiliations

  1. University of Coimbra, Coimbra, Portugal

    Luís Paquete

  2. Aberystwyth University, Aberystwyth, UK

    Christine Zarges

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Baioletti, M., Milani, A., Santucci, V. (2020). An Algebraic Approach for the Search Space of Permutations with Repetition. In: Paquete, L., Zarges, C. (eds) Evolutionary Computation in Combinatorial Optimization. EvoCOP 2020. Lecture Notes in Computer Science(), vol 12102. Springer, Cham. https://doi.org/10.1007/978-3-030-43680-3_2

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  • DOI: https://doi.org/10.1007/978-3-030-43680-3_2

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