{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T16:35:39Z","timestamp":1775666139511,"version":"3.50.1"},"reference-count":44,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,8]],"date-time":"2022-11-08T00:00:00Z","timestamp":1667865600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Deterioration of the searchability of Pareto dominance-based, many-objective evolutionary optimization algorithms is a well-known problem. Alternative solutions, such as scalarization-based and indicator-based approaches, have been proposed in the literature. However, Pareto dominance-based algorithms are still widely used. In this paper, we propose to redefine the calculation of Pareto-dominance. Instead of assigning solutions to non-dominated fronts, they are ranked according to the measure of dominating solutions referred to as k-Pareto optimality. In the case of probability measures, such re-definition results in an elegant and fast approximate procedure. Through experimental results on the many-objective 0\/1 knapsack problem, we demonstrate the advantages of the proposed approach: (1) the approximate calculation procedure is much faster than the standard sorting by Pareto dominance; (2) it allows for achieving higher hypervolume values for both multi-objective (two objectives) and many-objective (25 objectives) optimization; (3) in the case of many-objective optimization, the increased ability to differentiate between solutions results in a better compared to NSGA-II and NSGA-III. Apart from the numerical improvements, the probabilistic procedure can be considered as a linear extension of multidimentional topological sorting. It produces almost no ties and, as opposed to other popular linear extensions, has an intuitive interpretation.<\/jats:p>","DOI":"10.3390\/a15110420","type":"journal-article","created":{"date-parts":[[2022,11,9]],"date-time":"2022-11-09T02:34:52Z","timestamp":1667961292000},"page":"420","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["k-Pareto Optimality-Based Sorting with Maximization of Choice and Its Application to Genetic Optimization"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9514-1230","authenticated-orcid":false,"given":"Jean","family":"Ruppert","sequence":"first","affiliation":[{"name":"Mathematics and Computing S.\u00e0 r.l., L-2360 Luxembourg, Luxembourg"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1863-0129","authenticated-orcid":false,"given":"Marharyta","family":"Aleksandrova","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Faculty of Science, Technology and Medicine, University of Luxembourg, 2 Avenue de l\u2019Universite, L-4365 Esch-sur-Alzette, Luxembourg"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7374-3927","authenticated-orcid":false,"given":"Thomas","family":"Engel","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Faculty of Science, Technology and Medicine, University of Luxembourg, 2 Avenue de l\u2019Universite, L-4365 Esch-sur-Alzette, Luxembourg"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,8]]},"reference":[{"key":"ref_1","first-page":"7515","article-title":"A third generation genetic algorithm NSGAIII for task scheduling in cloud computing","volume":"34","author":"Imene","year":"2022","journal-title":"J. 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