Kernels of Mallows Models for Solving Permutation-based Problems

Recently, distance-based exponential probability models, such as Mallows and Generalized Mallows, have demonstrated their validity in the context of estimation of distribution algorithms (EDAs) for solving permutation problems. However, despite their successful performance, these models are unimodal, and therefore, they are not flexible enough to accurately model populations with solutions that are very sparse with regard to the distance metric considered under the model. In this paper, we propose using kernels of Mallows models under the Kendall's-tau and Cayley distances within EDAs. In order to demonstrate the validity of this new algorithm, Mallows Kernel EDA, we compare its performance with the classical Mallows and Generalized Mallows EDAs, on a benchmark of 90 instances of two different types of permutation problems: the quadratic assignment problem and the permutation flowshop scheduling problem. Experimental results reveal that, in most cases, Mallows Kernel EDA outperforms the Mallows and Generalized Mallows EDAs under the same distance. Moreover, the new algorithm under the Cayley distance obtains the best results for the two problems in terms of average fitness and computational time.

[1]  H. Mühlenbein,et al.  From Recombination of Genes to the Estimation of Distributions I. Binary Parameters , 1996, PPSN.

[2]  Jatinder N. D. Gupta,et al.  Flowshop scheduling research after five decades , 2006, Eur. J. Oper. Res..

[3]  Alexander Mendiburu,et al.  The Plackett-Luce ranking model on permutation-based optimization problems , 2013, 2013 IEEE Congress on Evolutionary Computation.

[4]  A. J. Clewett,et al.  Introduction to sequencing and scheduling , 1974 .

[5]  Alexander Mendiburu,et al.  Parallel EDAs to create multivariate calibration models for quantitative chemical applications , 2006, J. Parallel Distributed Comput..

[6]  J. A. Lozano,et al.  Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation , 2001 .

[7]  Joseph S. Verducci,et al.  Probability models on rankings. , 1991 .

[8]  Dirk Thierens,et al.  Multi-objective optimization with diversity preserving mixture-based iterated density estimation evolutionary algorithms , 2002, Int. J. Approx. Reason..

[9]  Pedro Larrañaga,et al.  Estimation of Distribution Algorithms , 2002, Genetic Algorithms and Evolutionary Computation.

[10]  Francisco Herrera,et al.  A study on the use of non-parametric tests for analyzing the evolutionary algorithms’ behaviour: a case study on the CEC’2005 Special Session on Real Parameter Optimization , 2009, J. Heuristics.

[11]  Martin Pelikan,et al.  An application of a multivariate estimation of distribution algorithm to cancer chemotherapy , 2008, GECCO '08.

[12]  Yi Mao,et al.  Non-parametric Modeling of Partially Ranked Data , 2007, NIPS.

[13]  T. Koopmans,et al.  Assignment Problems and the Location of Economic Activities , 1957 .

[14]  C. L. Mallows NON-NULL RANKING MODELS. I , 1957 .

[15]  Concha Bielza,et al.  Mateda-2.0: Estimation of Distribution Algorithms in MATLAB , 2010 .

[16]  Alexander Mendiburu,et al.  The linear ordering problem revisited , 2015, Eur. J. Oper. Res..

[17]  Bo Gao Estimation of Distribution Algorithms for Single- and Multi-Objective Optimization , 2014 .

[18]  Diana Carrera,et al.  Vine Estimation of Distribution Algorithms with Application to Molecular Docking , 2012 .

[19]  Pedro Larrañaga,et al.  Towards a New Evolutionary Computation - Advances in the Estimation of Distribution Algorithms , 2006, Towards a New Evolutionary Computation.

[20]  A. K. Ziver,et al.  Estimation of distribution algorithms for nuclear reactor fuel management optimisation , 2006 .

[21]  Pedro Larrañaga,et al.  Protein Folding in Simplified Models With Estimation of Distribution Algorithms , 2008, IEEE Transactions on Evolutionary Computation.

[22]  Siddhartha Shakya,et al.  Applications of Distribution Estimation Using Markov Network Modelling (DEUM) , 2012 .

[23]  Alexander Mendiburu,et al.  A review of distances for the Mallows and Generalized Mallows estimation of distribution algorithms , 2015, Comput. Optim. Appl..

[24]  Pedro Larrañaga,et al.  Globally Multimodal Problem Optimization Via an Estimation of Distribution Algorithm Based on Unsupervised Learning of Bayesian Networks , 2005, Evolutionary Computation.

[25]  Jing Liu,et al.  A survey of scheduling problems with setup times or costs , 2008, Eur. J. Oper. Res..

[26]  Pedro Larrañaga,et al.  Mixtures of Kikuchi Approximations , 2006, ECML.

[27]  Alexander Mendiburu,et al.  A review on estimation of distribution algorithms in permutation-based combinatorial optimization problems , 2012, Progress in Artificial Intelligence.

[28]  J. A. Lozano,et al.  Towards a New Evolutionary Computation: Advances on Estimation of Distribution Algorithms (Studies in Fuzziness and Soft Computing) , 2006 .

[29]  Thomas Brendan Murphy,et al.  Mixtures of distance-based models for ranking data , 2003, Comput. Stat. Data Anal..

[30]  Ekhine Irurozki,et al.  Sampling and learning the Mallows and Weighted Mallows models under the Hamming distance , 2014 .

[31]  Philip L. H. Yu,et al.  Mixtures of Weighted Distance-Based Models for Ranking Data , 2010, COMPSTAT.

[32]  Alexander Mendiburu,et al.  Introducing the Mallows Model on Estimation of Distribution Algorithms , 2011, ICONIP.

[33]  M. Fligner,et al.  Multistage Ranking Models , 1988 .

[34]  Alexander Mendiburu,et al.  Extending distance-based ranking models in estimation of distribution algorithms , 2014, 2014 IEEE Congress on Evolutionary Computation (CEC).

[35]  Josu Ceberio Solving permutation problems with estimation of distribution algorithms and extensions thereof , 2014 .

[36]  M. Fligner,et al.  Distance Based Ranking Models , 1986 .

[37]  David E. Goldberg,et al.  A Survey of Optimization by Building and Using Probabilistic Models , 2002, Comput. Optim. Appl..

[38]  Jose A. Lozano,et al.  Mallows model under the Ulam distance : a feasible combinatorial approach , 2014 .

[39]  Alexander Mendiburu,et al.  A Distance-Based Ranking Model Estimation of Distribution Algorithm for the Flowshop Scheduling Problem , 2014, IEEE Transactions on Evolutionary Computation.