scholarly journals Pareto-optimality and a search for robustness: choosing solutions with desired properties in objective space and parameter space

2011 ◽  
Vol 14 (2) ◽  
pp. 270-285 ◽  
Author(s):  
Gift Dumedah ◽  
Aaron A. Berg ◽  
Mark Wineberg
Keyword(s):  
Parameter Space ◽  
Assessment Tool ◽  
Pareto Optimal ◽  
Hydrological Models ◽  
Southern Ontario ◽  
Pareto Optimal Set ◽  
Creek Watershed ◽  
Objective Space ◽  
Optimal Set ◽  
Water Assessment

Multi-objective genetic algorithms are increasingly being applied to calibrate hydrological models by generating several competitive solutions usually referred to as a Pareto-optimal set. The Pareto-optimal set comprises non-dominated solutions at the calibration phase but it is usually unknown whether all or only a subset of non-dominated solutions at the calibration phase remains non-dominated at the validation phase. In practice, users would like to know solutions (and their associated properties) which remain non-dominated at both the calibration and validation phases. This study investigates robustness of the Pareto-optimal set by developing a model characterization framework (MCF). The MCF uses cluster analysis to examine the distribution of solutions in parameter space and objective space, and conditional probability to combine linkages between the distributions of solutions in both spaces. The MCF has been illustrated for calibration output generated from application of the Non-dominated Sorting Genetic Algorithm-II to calibrate the Soil and Water Assessment Tool for streamflow in the Fairchild Creek watershed in southern Ontario. Our results show that not all non-dominated solutions found at the calibration phase perform the same for different validation periods. The MCF illustrates that robust solutions – non-dominated solutions which cluster in similar locations in parameter space and objective space – performed consistently well for several validation periods.

scholarly journals Solution Selection from a Pareto Optimal Set of Multi-Objective Reservoir Operation via Clustering Operation Processes and Objective Values

Water ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1046
Author(s):  
Yanjun Kong ◽  
Yadong Mei ◽  
Xianxun Wang ◽  
Yue Ben
Keyword(s):  
Reservoir Operation ◽  
Pareto Optimal ◽  
Compromise Solution ◽  
Decision Space ◽  
Trade Off ◽  
Multi Objective ◽  
Pareto Optimal Set ◽  
Objective Space ◽  
Solution Selection ◽  
Optimal Set

Multi-objective evolutionary algorithms (MOEAs) are widely used to optimize multi-purpose reservoir operations. Considering that most outcomes of MOEAs are Pareto optimal sets with a large number of incomparable solutions, it is not a trivial task for decision-makers (DMs) to select a compromise solution for application purposes. Due to the increasing popularity of data-driven decision-making, we introduce a clustering-based decision-making method into the multi-objective reservoir operation optimization problem. Traditionally, solution selection has been conducted based on trade-off ranking in objective space, and solution characteristics in decision space have been ignored. In our work, reservoir operation processes were innovatively clustered into groups with unique properties in decision space, and the trade-off surfaces were analyzed via clustering in objective space. To attain a suitable performance, a new similarity measure, referred to as the Mei–Wang fluctuation similarity measure (MWFSM), was tailored to reservoir operation processes. This method describes time series in terms of both their shape and quantitative variation. Then, a compromise solution was selected via the joint use of two clustering results. A case study of the Three Gorges cascade reservoirs system under small and medium floods was investigated to verify the applicability of the proposed method. The results revealed that the MWFSM effectively distinguishes reservoir operation processes. Two more operation patterns with similar positions but different shapes were identified via MWFSM when compared with Euclidean distance and the dynamic time warping method. Furthermore, the proposed method decreased the selection range from the whole Pareto optimal set to a set containing relatively few solutions. Finally, a compromise solution was selected.

Analytical Derivation of the Pareto-Optimal Set with Application to Structural Design

2020 ◽  
pp. 43-66
Author(s):  
Federico Maria Ballo ◽  
Massimiliano Gobbi ◽  
Giampiero Mastinu ◽  
Giorgio Previati
Keyword(s):  
Structural Design ◽  
Pareto Optimal ◽  
Analytical Derivation ◽  
Pareto Optimal Set ◽  
Optimal Set

Optimal Approximation of Real Values Using Rational Numbers With Application to the Kinematic Design of Gearboxes

1993 ◽  
Author(s):  
Leonard P. Pomrehn ◽  
Panos Y. Papalambros
Keyword(s):  
Design Problem ◽  
Rational Numbers ◽  
Optimal Approximation ◽  
Pareto Optimal ◽  
Individual Stage ◽  
Gear Teeth ◽  
Kinematic Design ◽  
Pareto Optimal Set ◽  
Reducing Gear ◽  
Optimal Set

Abstract This article proposes a method for optimally approximating real values with rational numbers. Such requirements arise in the design of various types of gear sets, where integer numbers of gear teeth force individual stage ratios to assume rational values. The kinematic design of an 18-speed gearbox, taken from the literature, is analyzed and solved using the proposed method. The method, called sequential exhaustion, sequentially considers each stage of the gearbox design, exhaustively examining each stage. Examination of 94 solutions leads to a pareto-optimal set containing 11 solutions. Further, although the layout of the gearbox is predefined for the kinematic design problem, certain solutions of the problem exhibit “non-reducing” gear pairs, revealing previously unforeseen changes in the gearbox layout.

The Skewboid Method: A Simple and Effective Approach to Pareto Relaxation and Filtering

2012 ◽  
Author(s):  
Matthew I. Campbell
Keyword(s):  
Pareto Optimality ◽  
Population Based ◽  
Pareto Optimal ◽  
Large Set ◽  
Number Of Solutions ◽  
Target Number ◽  
Relaxation Methods ◽  
Pareto Optimal Set ◽  
Optimal Set ◽  
Adjustment Parameter

The concept of Pareto optimality is the default method for pruning a large set of candidate solutions in a multi-objective problem to a manageable, balanced, and rational set of solutions. While the Pareto optimality approach is simple and sound, it may select too many or too few solutions for the decision-maker’s needs or the needs of optimization process (e.g. the number of survivors selected in a population-based optimization). This inability to achieve a target number of solutions to keep has caused a number of researchers to devise methods to either remove some of the non-dominated solutions via Pareto filtering or to retain some dominated solutions via Pareto relaxation. Both filtering and relaxation methods tend to introduce many new adjustment parameters that a decision-maker (DM) must specify. In the presented Skewboid method, only a single parameter is defined for both relaxing the Pareto optimality condition (values between −1 and 0) and filtering more solutions from the Pareto optimal set (values between 0 and 1). This parameter can be correlated with a desired number of solutions so that this number of solutions is input instead of an unintuitive adjustment parameter. A mathematically sound derivation of the Skewboid method is presented followed by illustrative examples of its use. The paper concludes with a discussion of the method in comparison to similar methods in the literature.

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