Kernels of Mallows Models for Solving Permutation-based Problems

Abstract

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-τ 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.

An Analysis of Indirect Optimisation Strategies for Scheduling

“…MOSA essentially makes use of the ranks of solutions to direct the search and favours small genotype changes rather than larger ones such as those resulting from the use of crossover operators. This matches the principles of other algorithms coupling either mutation or probabilistic model sampling and selection mechanisms to move in the search space [7], [13]. xbest ← pop 1…”

RK-EDA: A Novel Random Key Based Estimation of Distribution Algorithm

“…They therefore struggle to produce competitive results [7]. Models that are more specific to permutations such as histogram models [16], [17], permutation distribution models [4], [6], [5] and factoradics [14] have shown better performances.…”

Mallows and generalized Mallows model for matchings

“…Recently, efficient algorithms for managing MM and GMM under the Cayley distance have been proposed [25]. The MM and GMM under the Cayley distance have already shown their utility in application problems such as the quadratic assignment problem (QAP) and the Permutation Flowshop Scheduling Problem (PFSP) [6] The Hamming distance is one of the most popular metrics for permutations [15], [28], [44]. Clearly, it is not a natural measure of disagreement between rankings, but in the case when permutations represent matchings of bipartite graphs, Hamming is the most reasonable choice.…”