A weight set decomposition algorithm for finding all efficient extreme points in the outcome set of a multiple objective linear program

Benson, Harold P.; Sun, Erjiang

doi:10.1016/s0377-2217(01)00153-9

Cited by 85 publications

(187 citation statements)

References 25 publications

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“…In the next section we review Benson's outer approximation algorithm [2,3], which has been shown to construct a set of ε-nondominated points for multi-objective linear programmes in [9].…”

Section: Convex Multi-objective Programming Problemmentioning

confidence: 99%

“…Benson has proposed an outer approximation algorithm (see [2,3]) to solve MOLPs in objective space, i.e.,…”

Section: Benson's Algorithmmentioning

confidence: 99%

“…Since the number of objectives of a multi-objective linear programming problem is often much smaller than the number of variables and typically many efficient solutions in decision space are mapped to a single nondominated point in objective space. Benson [3] argues that generating the nondominated set should require less computation than generating the efficient set. Moreover, it is reasonable to assume that a decision maker will choose a solution based on the objective values rather than variable values.…”

Section: Introductionmentioning

confidence: 99%

“…Benson has proposed an outer approximation method [3] to find the extended feasible set in objective space (and thereby the nondominated set) of an MOLP. In [15] and [9] some improvements for this algorithm have been suggested.…”

Section: Introductionmentioning

confidence: 99%

“…In Sect. 3 we review the version of Benson's algorithm given in [9] and prove a generalization of one of the results of [3]. Our algorithm for the convex case is given in Sect.…”

Section: Introductionmentioning

confidence: 99%

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An approximation algorithm for convex multi-objective programming problems

Shao

Ehrgott

2010

J Glob Optim

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In multi-objective convex optimization it is necessary to compute an infinite set of nondominated points. We propose a method for approximating the nondominated set of a multi-objective nonlinear programming problem, where the objective functions and the feasible set are convex. This method is an extension of Benson's outer approximation algorithm for multi-objective linear programming problems. We prove that this method provides a set of weakly ε-nondominated points. For the case that the objectives and constraints are differentiable, we describe an efficient way to carry out the main step of the algorithm, the construction of a hyperplane separating an exterior point from the feasible set in objective space. We provide examples that show that this cannot always be done in the same way in the case of non-differentiable objectives or constraints.

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“…In the next section we review Benson's outer approximation algorithm [2,3], which has been shown to construct a set of ε-nondominated points for multi-objective linear programmes in [9].…”

Section: Convex Multi-objective Programming Problemmentioning

confidence: 99%

“…Benson has proposed an outer approximation algorithm (see [2,3]) to solve MOLPs in objective space, i.e.,…”

Section: Benson's Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In Sect. 3 we review the version of Benson's algorithm given in [9] and prove a generalization of one of the results of [3]. Our algorithm for the convex case is given in Sect.…”

Section: Introductionmentioning

confidence: 99%

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An approximation algorithm for convex multi-objective programming problems

Shao

Ehrgott

2010

J Glob Optim

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Attribute Partitioning in Multiple Attribute Decision Making Problems for a Decision with a Purpose – a Fuzzy Approach

Gupta

Mohanty

2016

Multi Criteria Decision Anal

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This work introduces a methodology to find solutions corresponding to different purposes in a multiple attribute decisionmaking problem under fuzzy environment. The discernment of purpose-based solutions becomes important when the problem is defined vaguely and solution is targeted to heterogeneous population. Depending on the purpose, for which the solution is sought, the attributes are identified and weighted in an appropriate proportion. The level of similarity between a pair of attributes plays an important role to determine the aggregated value of attributes specific to a purpose. Our work determines the similarity levels between a pair of attributes by calculating their maximum attainability in presence of each other. The achievement of an attribute in presence of another is represented as a fuzzy set in the unit interval. The crisp equivalents of the fuzzy sets in the unit interval are used to define their simultaneous satisfaction denoted as 1-step relation. The 1-step relation is extended to (m-1)-step relation to calculate the degree of attainability of the same pair of attributes in the presence of m (all) attributes. The different levels of (m-1)-step relations generate several partitions of the attributes corresponding to multiple purposes in the multiple attribute decision-making problems. The degree of fulfilment of the purposes in the alternatives are numerically derived by first taking weighted average of attributes within the equivalence classes of a partition and then aggregating the values corresponding to equivalence classes through ordered weighted averaging. The methodology is illustrated with a numerical example.

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Output-sensitive complexity of multiobjective combinatorial optimization

Bökler

Ehrgott

Morris

et al. 2017

J. Multi-Crit. Decis. Anal.

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We study output-sensitive algorithms and complexity for multiobjective combinatorial optimization problems. In this computational complexity framework, an algorithm for a general enumeration problem is regarded efficient if it is output-sensitive, that is, its running time is bounded by a polynomial in the input and the output size. We provide both practical examples of multiobjective combinatorial optimization problems for which such an efficient algorithm exists as well as problems for which no efficient algorithm exists under mild complexity theoretic assumptions. KEYWORDScombinatorial optimization, linear programming, multiobjective optimization, output-sensitive complexity 1 1.1 gives us problems, which are not harder than a variant of the problem where we also want to find solutions. Hence, proving hardness of the problem as defined in Definition 1.1 will lead to a hardness result for the problem including finding of a representative solution.

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A weight set decomposition algorithm for finding all efficient extreme points in the outcome set of a multiple objective linear program

Cited by 85 publications

References 25 publications

An approximation algorithm for convex multi-objective programming problems

An approximation algorithm for convex multi-objective programming problems

Attribute Partitioning in Multiple Attribute Decision Making Problems for a Decision with a Purpose – a Fuzzy Approach

Output-sensitive complexity of multiobjective combinatorial optimization

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