Bound sets for biobjective combinatorial optimization problems

Ehrgott, Matthias; Gandibleux, Xavier

doi:10.1016/j.cor.2005.10.003

Cited by 99 publications

(77 citation statements)

References 16 publications

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“…To compute the reference set, we use the notion of ideal set [17], which is a lower bound of the Pareto front [10]. The ideal set is defined as the best potential Pareto front that can be produced from a minimal complete set of extreme supported efficient solutions.…”

Section: Reference Setmentioning

confidence: 99%

Speed-up techniques for solving large-scale biobjective TSP

Lust

Jaszkiewicz

2010

Computers & Operations Research

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Section: Reference Setmentioning

confidence: 99%

Speed-up techniques for solving large-scale biobjective TSP

Lust

Jaszkiewicz

2010

Computers & Operations Research

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“… Gandibleux and Klamroth () studied cardinality bounds for the MOKP based on weighted sum scalarizations. They showed that these bounds can be used to reduce the feasible set of the biobjective MOKP. Ehrgott and Gandibleux () introduced the concept of bound sets for MOCO problems. Indeed, well‐known bounds for multiobjective problems are the ideal point (lower bound) and the nadir point (upper bound) but first of all these bounds are not easy to compute, especially the nadir point, and second these values are very far from the Pareto front.…”

Section: The Multiobjective Knapsack Literaturementioning

confidence: 99%

“…r The average distance D 1 and maximal distance D 2 (to be minimized) (Czyzak and Jaszkiewicz, 1998;Ulungu et al, 1999) between the points of a reference set and the points of Z N , by using the Euclidean distance. Ideally, the reference set is Z N itself, but generally it is not available; otherwise, it can be the nondominated points existing among the union of various sets Z N generated by several methods, or an upper bound of Z N (Ehrgott and Gandibleux, 2007).…”

Section: Introductionmentioning

confidence: 99%

The multiobjective multidimensional knapsack problem: a survey and a new approach

Lust

Teghem

2012

Int Trans Operational Res

118

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The knapsack problem (KP) and its multidimensional version (MKP) are basic problems in combinatorial optimization. In this paper, we consider their multiobjective extension (MOKP and MOMKP), for which the aim is to obtain or approximate the set of efficient solutions. In the first step, we classify and briefly describe the existing works that are essentially based on the use of metaheuristics. In the second step, we propose the adaptation of the two‐phase Pareto local search (2PPLS) to the resolution of the MOMKP. With this aim, we use a very large scale neighborhood in the second phase of the method, that is the PLS. We compare our results with state‐of‐the‐art results and show that the results we obtained were never reached before by heuristics for biobjective instances. Finally, we consider the extension to three‐objective instances.

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“…A formulation of the vertex method appears in Pulkkinen (2006). For general concepts on bound sets, see Ehrgott and Gandibleux (2007).…”

Section: Preliminariesmentioning

confidence: 99%

A simple method for approximating a general Pareto surface

Lehtonen

Pulkkinen

2010

Multi Criteria Decision Anal

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We present a simple iterative procedure for approximating the Pareto surface of a set S and the only assumption is that S is closed and bounded. The algorithm creates a sequence of upper bounds for the Pareto surface and these upper bounds tighten towards the surface as the number of iteration loops increases towards infinity. The approximation method is further developed towards a tool for optimization in general situations and it may also be applied in interactive multiobjective optimization.

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Bound sets for biobjective combinatorial optimization problems

Cited by 99 publications

References 16 publications

Speed-up techniques for solving large-scale biobjective TSP

Speed-up techniques for solving large-scale biobjective TSP

The multiobjective multidimensional knapsack problem: a survey and a new approach

A simple method for approximating a general Pareto surface

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