Representation of the non-dominated set in biobjective discrete optimization

Vaz, Daniel; Paquete, Luís; Fonseca, Carlos M.; Klamroth, Kathrin; Stiglmayr, Michael

doi:10.1016/j.cor.2015.05.003

Cited by 26 publications

(22 citation statements)

References 21 publications

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“…To complement this hardness result, we notice that for discrete bi-objective optimisation problems Vaz et al (2015) provide algorithms that solve the discrete representation problem for known Y N in time polynomial in |Y N | and |R| for various combinations of coverage error, uniformity level, and ǫ-indicator as quality measures. Eusébio et al (2014) provide algorithms to compute δ-uniform or ǫ-representations for bi-objective integer network flow problems, but they do not analyse their complexity.…”

Section: Theorem 22 (mentioning

confidence: 99%

Discrete representation of non-dominated sets in multi-objective linear programming

Shao

Ehrgott

2016

European Journal of Operational Research

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In this paper we address the problem of representing the continuous but non-convex set of nondominated 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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Section: Theorem 22 (mentioning

confidence: 99%

Discrete representation of non-dominated sets in multi-objective linear programming

Shao

Ehrgott

2016

European Journal of Operational Research

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“…Thus, any representation problem for a biobjective discrete optimization problem using a combination of these quality measures as objectives can be solved in polynomial time by either dynamic programming or a threshold algorithm (for details, see Vaz et al . ).…”

Section: Bottleneck Objective Functionsmentioning

confidence: 97%

“…However, it is shown in Vaz et al . () that the uniformity problem is efficiently solvable for the nondominated points of a discrete biobjective problem.…”

Section: Bottleneck Objective Functionsmentioning

confidence: 99%

Easy to say they are Hard, but Hard to see they are Easy- Towards a Categorization of Tractable Multiobjective Combinatorial Optimization Problems

Figueira

Fonseca

Halffmann

et al. 2016

J. Multi-Crit. Decis. Anal.

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Multiobjective combinatorial optimization problems are known to be hard problems for two reasons: their decision versions are often NP-complete, and they are often intractable. Apart from this general observation, are there also variants or cases of multiobjective combinatorial optimization problems that are easy and, if so, what causes them to be easy? This article is a first attempt to provide an answer to these two questions. Thereby, a systematic description of reasons for easiness is envisaged rather than a mere collection of special cases. In particular, the borderline of easy and hard multiobjective optimization problems is explored.

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“…Faulkenberg et al, addressing a similar problem, use a bilevel programming formulation in which the upper level formulation controls spacing and the lower level formulation generates the nondominated solutions [18]. Other methods that focus on the bicriteria problem are due to Pereyra et al [37] and Vaz et al [47].…”

Section: Introductionmentioning

confidence: 99%

Bilevel programming for generating discrete representations in multiobjective optimization

Kirlik

Sayın

2017

Math. Program.

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The solution to a multiobjective optimization problem (MOP) consists of the nondominated set that portrays all relevant trade-off information. The ultimate goal is to identify a Decision Maker's most preferred solution without generating the entire set of nondominated solutions. We propose a bilevel programming formulation that can be used to this end. The bilevel program is capable of delivering an efficient solution that maps into a given set, provided that one exits. If the Decision Maker's preferences are known a priori, they can be used to specify the given set. Alternatively, we propose a method to obtain a representation of the nondominated set when the Decision Maker's preferences are not available. This requires a thorough search of the outcome space. The search can be facilitated by a partitioning scheme similar to the ones used in global optimization. Since the bilevel programming formulation either finds a nondominated solution in a given partition element or determines that there is none, a representation with a specified coverage error level can be found in a finite number of iterations. While building a discrete representation, the algorithm also generates an approximation of the nondominated set within the specified error factor. We illustrate the algorithm on the multiobjective linear programming problem.

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Representation of the non-dominated set in biobjective discrete optimization

Cited by 26 publications

References 21 publications

Discrete representation of non-dominated sets in multi-objective linear programming

Discrete representation of non-dominated sets in multi-objective linear programming

Easy to say they are Hard, but Hard to see they are Easy- Towards a Categorization of Tractable Multiobjective Combinatorial Optimization Problems

Bilevel programming for generating discrete representations in multiobjective optimization

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