On the Structure of the Non‐dominated Set for Bicriteria Programmes

TenHuisen, Matthew L.; Wiecek, Margaret M.

doi:10.1002/(sici)1099-1360(199609)5:3<232::aid-mcda103>3.3.co;2-z

Cited by 2 publications

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“…That is, we must systematically vary the values of f * i in the setX (33). The Bilevel Controlled Spacing Slicing Algorithm (BC-SSA) for general cones is shown in Fig.…”

Section: Slicing Methodsmentioning

confidence: 99%

“…The application of this definition to the nondominated set is straightforward. Definition 2.6 [32,33] A set, S ∈ R 2 , is disconnected, or separable by a hyperplane, if there exist a scalar, α, and a hyperplane, H , of the form (i) H := {y ∈ R 2 : y 1 = α}, or (ii) H := {y ∈ R 2 : y 2 = α}, and the following properties hold: H ∩ S = ∅, H ≤ ∩ S = ∅, and H ≥ ∩ S = ∅. Here, if H has the form of (i), H ≤ := {y ∈ R 2 : y 1 ≤ α} and H ≥ := {y ∈ R 2 : y 1 ≥ α}.…”

Section: Problem Formulation and Preliminariesmentioning

confidence: 99%

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Generating equidistant representations in biobjective programming

Faulkenberg

Wiecek

2011

Comput Optim Appl

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In recent years, emphasis has been placed on generating quality representations of the nondominated set of multiobjective optimization problems. This paper presents two methods for generating discrete representations with equidistant points for biobjective problems with solution sets determined by convex, polyhedral cones. The Constraint Controlled-Spacing method is based on the epsilon-constraint method with an additional constraint to control the spacing of generated points. The Bilevel Controlled-Spacing method has a bilevel structure with the lower-level generating the nondominated points and the upper-level controlling the spacing, and is extended to multiobjective problems. Both methods are proven to produce (weakly) nondominated points and are demonstrated on a variety of test problems.

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“…That is, we must systematically vary the values of f * i in the setX (33). The Bilevel Controlled Spacing Slicing Algorithm (BC-SSA) for general cones is shown in Fig.…”

Section: Slicing Methodsmentioning

confidence: 99%

Section: Problem Formulation and Preliminariesmentioning

confidence: 99%

Generating equidistant representations in biobjective programming

Faulkenberg

Wiecek

2011

Comput Optim Appl

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“…composed of two curves with empty intersection. A connected set was defined by Bitran andMagnanti (1979), while TenHuisen andWiecek (1996) applied that concept to define the disconnected non-dominated set for BCPs and whose definition we use.…”

Section: Problem Formulationmentioning

confidence: 99%

Piecewise quadratic approximation of the non‐dominated set for bi‐criteria programs

Wiecek

Chen

Zhang

2001

Multi Criteria Decision Anal

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A procedure to approximate the non-dominated set for general (continuous) bi-criteria programs is proposed. The piecewise approximation is composed of quadratic curves, each of which is developed locally in a neighbourhood of a non-dominated point of interest and based on primal-dual relationships associated with the weighted Tchebycheff scalarization of the original problem. The approximating quadratic functions, in which decision maker's preferences are represented, give a closed-form description of the non-dominated set. A numerical example is included.

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On the Structure of the Non‐dominated Set for Bicriteria Programmes

Cited by 2 publications

References 0 publications

Generating equidistant representations in biobjective programming

Generating equidistant representations in biobjective programming

Piecewise quadratic approximation of the non‐dominated set for bi‐criteria programs

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