Uncertainty quantifications of Pareto optima in multiobjective problems

Hung, Tzu-Chieh; Chan, Kuei-Yuan

doi:10.1007/s10845-011-0602-9

Cited by 12 publications

(7 citation statements)

References 14 publications

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“…A methodology to calculate the so called optimality influence range of the Pareto approximation due to variations in the design space has been proposed in [14]. The optimality influence range is a hyper-rectangle that encloses all the objective variations with an angle.…”

Section: Multiobjective Optimization Methodsmentioning

confidence: 99%

“…The incumbent Pareto Front is updated comparing the initial incumbent Pareto Front and the 9 additional runs. The new Pareto solutions are 1,4,7,8,12,14,15,16,18,19,21,22,23, and 24.…”

Section: Update Incumbent Pareto Frontmentioning

confidence: 99%

“…The corresponding input and output values are shown on Table 3 (runs 25-33). Afterwards, the incumbent Pareto front was updated and the new Pareto solutions are 1,4,7,8,12,14,15,16,18,19,21,22,23,24,[27][28][29][30][31][32][33]. Then, the stopping criteria were evaluated and since the R 2 of both models are larger than 0.98 the method stopped and the final Pareto solutions are reported.…”

Section: Evaluate Stopping Criteriamentioning

confidence: 99%

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Simulation and multi-objective optimization to improve the final shape and process efficiency of a laser-based material accumulation process

et al. 2020

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Common goals of modern production processes are precision and efficiency. Typically, they are conflicting and cannot be optimized at the same time. Multi-objective optimization methods are able to compute a set of good parameters, from which a decision maker can make a choice for practical situations. For complex processes, the use of physical experiments and/or extensive process simulations can be too costly or even unfeasible, so the use of surrogate models based on few simulations is a good alternative. In this work, we present an integrated framework to find optimal process parameters for a laser-based material accumulation process (thermal upsetting) using a combination of meta-heuristic optimization models and finite element simulations. In order to effectively simulate the coupled system of heat equation with solid-liquid phase transitions and melt flow with capillary free surface in three space dimensions for a wide range of process parameters, we introduce a new coupled numerical 3d finite element method. We use a multi-objective optimization method based on surrogate models. Thus, with only few direct simulations necessary, we are able to select Pareto sets of process parameters which can be used to optimize three or six different performance measures.

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Section: Multiobjective Optimization Methodsmentioning

confidence: 99%

“…The incumbent Pareto Front is updated comparing the initial incumbent Pareto Front and the 9 additional runs. The new Pareto solutions are 1,4,7,8,12,14,15,16,18,19,21,22,23, and 24.…”

Section: Update Incumbent Pareto Frontmentioning

confidence: 99%

Section: Evaluate Stopping Criteriamentioning

confidence: 99%

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Simulation and multi-objective optimization to improve the final shape and process efficiency of a laser-based material accumulation process

et al. 2020

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“…Existing works mainly focus on the small variations in DVs and DEPs [3,4,5,6,7,8,9,10,1,11,12,13,14,15,16,17,18,19]. Moreover, some previous work focused on the variations in the performance function model [20,21,22,23].…”

Section: Nomenclaturementioning

confidence: 99%

“…Optimal design is primary and robust design is secondary: It is one post-optimality approach. The final solution is selected from the Pareto optimal set based on the robustness criterion [68,69,70,24,25,71,18,19,72]. 3.…”

Section: Multi-objective Robust Optimization Problemmentioning

confidence: 99%

Multi-Objective Robust Optimization Using a Postoptimality Sensitivity Analysis Technique: Application to a Wind Turbine Design

Caro¹,

Bennis

Soto

et al. 2015

Journal of Mechanical Design

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Toward a multi-objective optimization robust problem, the variations in design variables and design environment parameters include the small variations and the large variations. The former have small effect on the performance functions and/or the constraints, and the latter refer to the ones that have large effect on the performance functions and/or the constraints. The robustness of performance functions is discussed in this paper. A post-optimality sensitivity analysis technique for multi-objective robust optimization problems is discussed and two robustness indices are introduced. The first one considers the robustness of the performance functions to small variations in the design variables and the design environment parameters. The second robustness index characterizes the robustness of the performance functions to large variations in the design environment parameters. It is based on the ability of a solution to maintain a good Pareto ranking for different design environment parameters due to large variations. The robustness of the solutions is treated as vectors in the robustness function space, which is defined by the two proposed robustness indices. As a result, the designer can compare the robustness of all Pareto optimal solutions and make a decision. Finally, two illustrative examples are given to highlight the contributions of this paper. The first example is about a numerical problem, whereas the second problem deals with the multi-objective robust optimization design of a floating wind turbine. * Corresponding author: Email: stephane.caro@irccyn.ec-nantes.fr, Tel: +33 (0)2 40 37 69 68, Fax: +33 (0)2 40 37 69 30 Nomenclature x design variable vector p vector of design environment parameters g k the kth constraint f performance function vector F feasible set P Pareto optimal set σ standard deviation µ expected value I RS robustness index against small variations I F feasibility index of a solution I P Pareto optimality index of a solution h(p) probability density function of p I rank individual's ranking N number of discrete values of design environment parameters I RL robustness index against large variations P R (P) the most robust solutions amongst the Pareto optimal solutions P produced power of wind turbine rotor F a the thrust force in the partial load region of a wind turbine rotor γ r the root twist angle γ t the tip twist angle c r the chord length at the root c t the chord length at the tip ω rotor rotational speed r r root radius of the wind turbine IntroductionMany design optimization problems are Multi-Objective Optimization Problems (MOOP) and are subject to uncertainties or variations in their parameters. Robustness is a product's ability to maintain its performance under the variations in its parameters. Robust design aims at maximizing product's robustness. In other words, it aims at minimizing the sensitivity of performance to variations without controlling the source of these variations [1]. Sometimes, the robustness of a product is as important as or even more important than its p...

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Set‐Based Design

Dullen,

Verma

2023

Systems Engineering for the Digital Age

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Uncertainty quantifications of Pareto optima in multiobjective problems

Cited by 12 publications

References 14 publications

Simulation and multi-objective optimization to improve the final shape and process efficiency of a laser-based material accumulation process

Simulation and multi-objective optimization to improve the final shape and process efficiency of a laser-based material accumulation process

Multi-Objective Robust Optimization Using a Postoptimality Sensitivity Analysis Technique: Application to a Wind Turbine Design

Set‐Based Design

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