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Hybrid multi-objective Bayesian estimation of distribution algorithm: a comparative analysis for the multi-objective knapsack problem

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Abstract

Nowadays, a number of metaheuristics have been developed for efficiently solving multi-objective optimization problems. Estimation of distribution algorithms are a special class of metaheuristic that intensively apply probabilistic modeling and, as well as local search methods, are widely used to make the search more efficient. In this paper, we apply a Hybrid Multi-objective Bayesian Estimation of Distribution Algorithm (HMOBEDA) in multi and many objective scenarios by modeling the joint probability of decision variables, objectives, and the configuration parameters of an embedded local search (LS). We analyze the benefits of the online configuration of LS parameters by comparing the proposed approach with LS off-line versions using instances of the multi-objective knapsack problem with two to five and eight objectives. HMOBEDA is also compared with five advanced evolutionary methods using the same instances. Results show that HMOBEDA outperforms the other approaches including those with off-line configuration. HMOBEDA not only provides the best value for hypervolume indicator and IGD metric in most of the cases, but it also computes a very diverse solutions set close to the estimated Pareto front.

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Notes

  1. The discretization process converts each objective value into \(sd_r\) discrete states considering the maximum possible value for each objective (\(Max_r\)). For each objective r, its discrete value is calculated as \( zd_r = \left\lceil z_r sd_r / Max_r \right\rceil \).

  2. This approach is used in an usual application for MOKP (Zhang and Li 2007), which can be downloaded from http://http://dces.essex.ac.uk/staff/zhang/webofmoead.htm and is also suggested in Ishibuchi et al. (2015).

  3. This implementation is used in Bader and Zitzler (2011) for more than three objectives and can be downloaded from http://www.tik.ee.ethz.ch/sop/download/supplementary/hype/.

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Acknowledgements

M. Delgado acknowledges CNPq Grant 309197/2014-7. M. Martins acknowledges CAPES/Brazil. R. Santana acknowledges support by: IT-609-13 Program (Basque Government), TIN2013-41272P (Spanish Ministry of Science and Innovation) and CNPq Program Science Without Borders No. 400125/2014-5 (Brazil Government).

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  1. Federal University of Technology - Paraná (UTFPR), Av. Sete de Setembro, 3165, Curitiba, PR, Brazil

    Marcella S. R. Martins, Myriam R. B. S. Delgado & Ricardo Lüders

  2. University of the Basque Country (UPV/EHU), 20018, Donostia, San Sebastián, Spain

    Roberto Santana

  3. Midwest State University of Parana (UNICENTRO), Guarapuava, PR, Brazil

    Richard A. Gonçalves & Carolina P. de Almeida

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  1. Marcella S. R. Martins
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Martins, M.S.R., Delgado, M.R.B.S., Lüders, R. et al. Hybrid multi-objective Bayesian estimation of distribution algorithm: a comparative analysis for the multi-objective knapsack problem. J Heuristics 24, 25–47 (2018). https://doi.org/10.1007/s10732-017-9356-7

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