A Recursive Algorithm for Finding All Nondominated Extreme Points in the Outcome Set of a Multiobjective Integer Programme

Przybylski, Anthony; Gandibleux, Xavier; Ehrgott, Matthias

doi:10.1287/ijoc.1090.0342

Cited by 80 publications

(69 citation statements)

References 13 publications

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“…A reason why this work has not yet been realized is that the first phase of the 2PPLS with three or more objective is more complicated, because the correspondance between the geometrical and the algebraic interpretation of the weight sets is not more valid (Przybylski et al 2007). Experiments should be done to see if the use of a sophisticated method as first phase will be yet worthwhile, or if a simple method based on randomly or uniformly generated weight sets would be sufficient.…”

Section: Resultsmentioning

confidence: 99%

Two-phase Pareto local search for the biobjective traveling salesman problem

2009

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In this work, we present a method, called Two-Phase Pareto Local Search, to find a good approximation of the efficient set of the biobjective traveling salesman problem. In the first phase of the method, an initial population composed of a good approximation of the extreme supported efficient solutions is generated. We use as second phase a Pareto Local Search method applied to each solution of the initial population. We show that using the combination of these two techniques: good initial population generation plus Pareto Local Search gives better results than stateof-the-art algorithms. Two other points are introduced: the notion of ideal set and a simple way to produce near-efficient solutions of multiobjective problems, by using an efficient single-objective solver with a data perturbation technique.

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Section: Resultsmentioning

confidence: 99%

Two-phase Pareto local search for the biobjective traveling salesman problem

2009

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“…In a more general case of a linear but unknown utility function, it is sufficient to identify all extreme supported nondominated objective vectors, since the optimal solution must come from this set. Przybylski et al [2010b], Özpeynirci and Köksalan [2010] propose similar algorithms to identify all extreme nondominated objective vectors for the MOIP problem. Both algorithms use a weighted single objective function and partition the weight space in order to enumerate the extreme supported nondominated set; an approach first proposed for Multi-Objective Linear Programming (MOLP) by Sun [2000, 2002].…”

Section: Literaturementioning

confidence: 99%

Multi-Objective Integer Programming: An Improved Recursive Algorithm

Özlen

Burton

MacRae

2013

J Optim Theory Appl

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This paper introduces an improved recursive algorithm to generate the set of all nondominated objective vectors for the Multi-Objective Integer Programming (MOIP) problem. We significantly improve the earlier recursive algorithm of Özlen and Azizoğlu by using the set of already solved subproblems and their solutions to avoid solving a large number of IPs. A numerical example is presented to explain the workings of the algorithm, and we conduct a series of computational experiments to show the savings that can be obtained. As our experiments show, the improvement becomes more significant as the problems grow larger in terms of the number of objectives.

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“…Among these, the Evans-Steuer algorithm [15] is implemented as a software called ADBASE [31]. The idea of decomposing the parameter set is also used to solve multiobjective integer programs, see for instance [26].In [27], Ruszczyński and Vanderbei developed an algorithm to solve LVOPs with two objectives and the efficiency of this algorithm is equivalent to solving a single scalar linear program by the parametric simplex algorithm. Indeed, the algorithm is a modification of the parametric simplex method and it produces a subset of efficient solutions that generate the whole efficient frontier in case the problem is bounded.Apart from the algorithms that work in the variable or parameter space, there are algorithms working in the objective space.…”

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confidence: 99%

A parametric simplex algorithm for linear vector optimization problems

Rudloff¹,

Ulus²,

Vanderbei³

2016

Math. Program.

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In this paper, a parametric simplex algorithm for solving linear vector optimization problems (LVOPs) is presented. This algorithm can be seen as a variant of the multi-objective simplex (the Evans-Steuer) algorithm [15]. Different from it, the proposed algorithm works in the parameter space and does not aim to find the set of all efficient solutions. Instead, it finds a solution in the sense of Löhne [19], that is, it finds a subset of efficient solutions that allows to generate the whole efficient frontier. In that sense, it can also be seen as a generalization of the parametric self-dual simplex algorithm, which originally is designed for solving single objective linear optimization problems, and is modified to solve two objective bounded LVOPs with the positive orthant as the ordering cone in Ruszczyński and Vanderbei [27]. The algorithm proposed here works for any dimension, any solid pointed polyhedral ordering cone C and for bounded as well as unbounded problems.Numerical results are provided to compare the proposed algorithm with an objective space based LVOP algorithm (Benson's algorithm in [16]), that also provides a solution in the sense of [19], and with the Evans-Steuer algorithm [15]. The results show that for nondegenerate problems the proposed algorithm outperforms Benson's algorithm and is on par with the Evan-Steuer algorithm. For highly degenerate problems Benson's algorithm [16] outperforms the simplex-type algorithms; however, the parametric simplex algorithm is for these problems computationally much more efficient than the Evans-Steuer algorithm.

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A Recursive Algorithm for Finding All Nondominated Extreme Points in the Outcome Set of a Multiobjective Integer Programme

Cited by 80 publications

References 13 publications

Two-phase Pareto local search for the biobjective traveling salesman problem

Two-phase Pareto local search for the biobjective traveling salesman problem

Multi-Objective Integer Programming: An Improved Recursive Algorithm

A parametric simplex algorithm for linear vector optimization problems

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