Discrete representation of non-dominated sets in multi-objective linear programming
•Discrete representation of non-dominated sets relates to NP-hard location problems.•Known methods do not have the coverage property or bound the uniformity level.•Under some assumptions the RNBI method has bounded uniformity level and coverage error.•Empirical tests on applied problems and random e...
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| Vydáno v: | European journal of operational research Ročník 255; číslo 3; s. 687 - 698 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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Amsterdam
Elsevier B.V
16.12.2016
Elsevier Sequoia S.A |
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| ISSN: | 0377-2217, 1872-6860 |
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| Abstract | •Discrete representation of non-dominated sets relates to NP-hard location problems.•Known methods do not have the coverage property or bound the uniformity level.•Under some assumptions the RNBI method has bounded uniformity level and coverage error.•Empirical tests on applied problems and random examples validate the RNBI method.
In this paper we address the problem of representing the continuous but non-convex set of non-dominated points of a multi-objective linear programme by a finite subset of such points. We prove that a related decision problem is NP-complete. Moreover, we illustrate the drawbacks of the known global shooting, normal boundary intersection and normal constraint methods concerning the coverage error and uniformity level of the representation by examples. We propose a method which combines the global shooting and normal boundary intersection methods. By doing so, we overcome their limitations, but preserve their advantages. We prove that our method computes a set of evenly distributed non-dominated points for which the coverage error and the uniformity level can be guaranteed. We apply this method to an optimisation problem in radiation therapy and present illustrative results for some clinical cases. Finally, we present numerical results on randomly generated examples. |
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| AbstractList | •Discrete representation of non-dominated sets relates to NP-hard location problems.•Known methods do not have the coverage property or bound the uniformity level.•Under some assumptions the RNBI method has bounded uniformity level and coverage error.•Empirical tests on applied problems and random examples validate the RNBI method.
In this paper we address the problem of representing the continuous but non-convex set of non-dominated points of a multi-objective linear programme by a finite subset of such points. We prove that a related decision problem is NP-complete. Moreover, we illustrate the drawbacks of the known global shooting, normal boundary intersection and normal constraint methods concerning the coverage error and uniformity level of the representation by examples. We propose a method which combines the global shooting and normal boundary intersection methods. By doing so, we overcome their limitations, but preserve their advantages. We prove that our method computes a set of evenly distributed non-dominated points for which the coverage error and the uniformity level can be guaranteed. We apply this method to an optimisation problem in radiation therapy and present illustrative results for some clinical cases. Finally, we present numerical results on randomly generated examples. In this paper we address the problem of representing the continuous but non-convex set of non-dominated points of a multi-objective linear programme by a finite subset of such points. We prove that a related decision problem is NP-complete. Moreover, we illustrate the drawbacks of the known global shooting, normal boundary intersection and normal constraint methods concerning the coverage error and uniformity level of the representation by examples. We propose a method which combines the global shooting and normal boundary intersection methods. By doing so, we overcome their limitations, but preserve their advantages. We prove that our method computes a set of evenly distributed non-dominated points for which the coverage error and the uniformity level can be guaranteed. We apply this method to an optimisation problem in radiation therapy and present illustrative results for some clinical cases. Finally, we present numerical results on randomly generated examples. |
| Author | Ehrgott, Matthias Shao, Lizhen |
| Author_xml | – sequence: 1 givenname: Lizhen surname: Shao fullname: Shao, Lizhen email: lshao@ustb.edu.cn organization: Key Laboratory of Advanced Control of Iron and Steel Process (Ministry of Education), School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China – sequence: 2 givenname: Matthias surname: Ehrgott fullname: Ehrgott, Matthias email: m.ehrgott@lancaster.ac.uk organization: Department of Management Science, Lancaster University Management School, Bailrigg, Lancaster LA1 4YX, United Kingdom |
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| Cites_doi | 10.1016/j.cor.2014.02.009 10.1007/s00453-001-0022-x 10.1007/s10957-005-5494-4 10.1007/s10479-009-0659-4 10.1137/S1052623496307510 10.1023/A:1008215702611 10.1016/j.cor.2015.05.003 10.2514/1.8977 10.1007/s00158-007-0185-4 10.1016/0305-0548(91)90040-X 10.1007/s00158-013-0925-6 10.1007/s00158-006-0071-5 10.1002/(SICI)1520-6750(199702)44:1<47::AID-NAV3>3.0.CO;2-M 10.1007/s00158-011-0729-5 10.1016/0022-247X(75)90189-4 10.1007/BF00932907 10.1007/s101070050128 10.1287/opre.51.3.427.14951 10.1007/s11081-009-9099-x 10.1007/s00158-011-0698-8 10.1007/s00158-002-0276-1 10.1007/s00186-008-0220-2 10.1007/s10589-011-9403-5 10.1287/opre.22.1.189 10.1137/080729013 |
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| Snippet | •Discrete representation of non-dominated sets relates to NP-hard location problems.•Known methods do not have the coverage property or bound the uniformity... In this paper we address the problem of representing the continuous but non-convex set of non-dominated points of a multi-objective linear programme by a... |
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| SubjectTerms | Decision making Discrete representation Linear programming Multi-objective optimisation Non-dominated set Optimization techniques Problem solving Radiation therapy Studies |
| Title | Discrete representation of non-dominated sets in multi-objective linear programming |
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