G–Indicator: An M–Ary Quality Indicator for the Evaluation of Non–dominated Sets

Lizárraga, Giovanni; Botello, Salvador

doi:10.1007/978-3-540-76631-5_12

Cited by 5 publications

(7 citation statements)

References 4 publications

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“…The g-Indicator (G), which was proposed by Lizárraga et al (2008), is a metric to evaluate the diversity and distribution of solutions on the approximated front. In its generalized form, the metric is defined by the union of the hyper-volume covered by hyper-spheres of radius U centered at each point of the Pareto front approximation.…”

Section: Comparison Of Metricsmentioning

confidence: 99%

Improved multiobjective differential evolution with spherical pruning algorithm for optimizing 3D printing technology parametrization process

et al. 2021

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Multiobjective optimization approaches have allowed the improvement of technical features in industrial processes, focusing on more accurate approaches for solving complex engineering problems and support decision-making. This paper proposes a hybrid approach to optimize the 3D printing technology parameters, integrating the design of experiments and multiobjective optimization methods, as an alternative to classical parametrization design used in machining processes. Alongside the approach, a multiobjective differential evolution with uniform spherical pruning (usp-MODE) algorithm is proposed to serve as an optimization tool. The parametrization design problem considered in this research has the following three objectives: to minimize both surface roughness and dimensional accuracy while maximizing the mechanical resistance of the prototype. A benchmark with nondominated sorting genetic algorithm II (NSGA-II) and with the classical sp-MODE is used to evaluate the performance of the proposed algorithm. With the increasing complexity of engineering problems and advances in 3D printing technology, this study demonstrates the applicability of the proposed hybrid approach, finding optimal combinations for the machining process among conflicting objectives regardless of the number of decision variables and goals involved. To measure the performance and to compare the results of metaheuristics used in this study, three Pareto comparison metrics have been utilized to evaluate both the convergence and diversity of the obtained Pareto approximations for each algorithm: hyper-volume (H), g-Indicator (G), and inverted generational distance (IGD). To all of them, ups-MODE outperformed, with significant figures, the results reached by NSGA-II and sp-MODE algorithms.

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Section: Comparison Of Metricsmentioning

confidence: 99%

Improved multiobjective differential evolution with spherical pruning algorithm for optimizing 3D printing technology parametrization process

et al. 2021

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“…A precision of 0.001 was set for each variable in the phenotype. The three algorithms were run 30 times with each mating-mutation configuration, the average behavior of each configuration was assessed using a version of G − metric [14] to work in Ï, a n-ary quality indicator that ranks P F * known s based on the their attained dispersion and convergence.…”

Section: Testing Rankmoeamentioning

confidence: 99%

A RankMOEA to approximate the pareto front of a dynamic principal-agent model

Ortiz

Rodríguez-Vázquez

Cabral

et al. 2011

Proceedings of the 13th Annual Conference on Genetic and Evolutionary Computation

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In this paper, a new Multi-Objective Evolutionary Algorithm (MOEA) named RankMOEA is proposed. Innovative niching and ranking-mutation procedures which avoid the need of parameters definition are involved; such procedures outperform traditional diversity-preservation mechanisms under spread-hardness situations. RankMOEA performance is compared with those of other state of the art MOEAs: MOGA, NSGA-II and SPEA2, showing remarkable improvements. RankMOEA is also applied to approximate the Pareto Front of a Dynamic Principal-Agent model with Discrete Actions posed in a Multi-Objective Optimization framework allowing to consider more powerful assumptions than those used in the traditional single-objective optimization approach. Within this new framework a set of feasible contracts is described, while others similar studies only focus on one single contract. The results achieved with RankMOEA show better spread and minor error than those obtained by already mentioned MOEAs, allowing to perform better economic analysis in the contracts trade-off surface.

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“…Since most MCO algorithms have historically operated by iteratively optimizing the underlying MCO problem to exactly sample the Pareto surface, there has been little research into interpolated surface similarity metrics. Instead, most previous surface comparison research has focused on evaluating the relative performance between different Pareto surface interpolations, that is, which of the interpolated surfaces is superior 14–22 . This type of evaluation is not as useful for determining the numerical similarity between the interpolated surfaces, which is more relevant when creating machine learning MCO models aimed at exactly replicating the results of a different MCO algorithm.…”

Section: Introductionmentioning

confidence: 99%

“…Instead, most previous surface comparison research has focused on evaluating the relative performance between different Pareto surface interpolations, that is, which of the interpolated surfaces is superior. [14][15][16][17][18][19][20][21][22] This type of evaluation is not as useful for determining the numerical similarity between the interpolated surfaces, which is more relevant when creating machine learning MCO models aimed at exactly replicating the results of a different MCO algorithm. Particularly in radiation therapy MCO, the vertices on the surface are linearly interpolated to infer possible dose distributions and trade-offs, and most of the previously developed surface comparison techniques do not take this into account.…”

Section: Introductionmentioning

confidence: 99%

Technical note: Interpolated Pareto surface similarity metrics for multi‐criteria optimization in radiation therapy

Jensen

Zhang

2020

Medical Physics

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Purpose There is a strong clinical need to evaluate different multi‐criteria optimization (MCO) algorithms, including inverse optimization sampling algorithms and machine learning‐based predictions. This study aims to develop and compare several interpolated Pareto surface similarity metrics. Materials and methods The first metric is the root‐mean‐square error (RMSE) evaluated between vertices on the interpolated surfaces, augmented by intra‐simplex sampling of the barycentric coordinates of the surfaces’ simplicial complexes. The second metric is the average projected distance (APD), which evaluates the displacements between the vertices and computes their projections along the mean displacement. The third metric is the average nearest‐point distance (ANPD), which numerically integrates point‐to‐simplex distances over the sampled simplices of the interpolated surfaces. These metrics were compared by their convergence rates, the times required to achieve convergence, and their representation of the underlying surface interpolations. For analysis, several interpolated Pareto surface pairs were constructed abstractly, with one pair from a nasopharyngeal treatment planning case using MCO. Results Convergence within 1% is typically achieved at approximately 50 and 80 samples per barycentric dimension for the RMSE and the ANPD, respectively. Calculation requires approximately 1 and 10 ms to achieve convergence for the RMSE and the ANPD in two dimensions, respectively, while the APD always requires < 1 ms. These time costs are much higher in higher dimensions for just the RMSE and ANPD. The APD values more closely approximated the ANPD limits than the RMSE limits. Conclusion The ANPD’s formulation and generality make it likely more meaningful than the RMSE and APD for representing the similarity between the underlying interpolated surfaces rather than the sampling points on the surfaces. However, in situations requiring high‐speed evaluations, the APD may be more desirable due to its speed, independence from a subjectively chosen sampling rate, and similarity to the ANPD limits.

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G–Indicator: An M–Ary Quality Indicator for the Evaluation of Non–dominated Sets

Cited by 5 publications

References 4 publications

Improved multiobjective differential evolution with spherical pruning algorithm for optimizing 3D printing technology parametrization process

Improved multiobjective differential evolution with spherical pruning algorithm for optimizing 3D printing technology parametrization process

A RankMOEA to approximate the pareto front of a dynamic principal-agent model

Technical note: Interpolated Pareto surface similarity metrics for multi‐criteria optimization in radiation therapy

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