A new graphical visualization of n-dimensional Pareto front for decision-making in multiobjective optimization

Abstract

a b s t r a c tNew challenges in engineering design lead to multiobjective (multicriteria) problems. In this context, the Pareto front supplies a set of solutions where the designer (decision-maker) has to look for the best choice according to his preferences. Visualization techniques often play a key role in helping decision-makers, but they have important restrictions for more than two-dimensional Pareto fronts. In this work, a new graphical representation, called Level Diagrams, for n-dimensional Pareto front analysis is proposed. Level Diagrams consists of representing each objective and design parameter on separate diagrams. This new technique is based on two key points: classification of Pareto front points according to their proximity to ideal points measured with a specific norm of normalized objectives (several norms can be used); and synchronization of objective and parameter diagrams. Some of the new possibilities for analyzing Pareto fronts are shown. Additionally, in order to introduce designer preferences, Level Diagrams can be coloured, so establishing a visual representation of preferences that can help the decision-maker. Finally, an example of a robust control design is presented -a benchmark proposed at the American Control Conference. This design is set as a six-dimensional multiobjective problem.Ó 2008 Elsevier Inc. All rights reserved. MotivationIn numerous engineering areas, the task of obtaining suitable designs becomes a multiobjective (or multicriteria) problem. This means it is necessary to look for a solution in the design space that satisfies several specifications (objectives) in the performance space. Generally, these specifications are conflicting, that is, there is no simultaneous optimal solution for all of them. In this context, the solution is not unique, instead there is a set of possible solutions where none is best for all objectives. This set of optimal solutions in the design space is called the Pareto set. The region defined by the performances (the value of all objectives) for all Pareto set points is called the Pareto front.The exact determination of the Pareto front is unrealistic for real-world problems, as it is usually an infinite set. Therefore, it is usual to focus on obtaining a discrete approximation. A common step for solving a multiobjective optimization problem is to obtain the discrete approximation of the Pareto front. This is an open research field where numerous techniques have already been developed [19] and where new techniques are being constantly developed [17,14]. An alternative, and very active research line, is Multiobjective Evolutionary Algorithms [5,9]. In general, these algorithms supply reasonable solutions for Pareto front approximations. Once obtained, the next step for the designer is to select one, or more, solutions inside the Pareto front approximation. The final solution is often selected using methodologies that normally include designer 0020-0255/$ -see front matter Ó

Multiobjective sensitivity analysis and optimization of distributed hydrologic model MOBIDIC

“…To analyze the Pareto solution and also to compare it with the solution from SOO, except for traditional methods, the "level diagram" proposed by Blasco et al (2008) was also used. Compared to traditional methods, it can visualize highdimensional Pareto fronts, and synchronizes the objective and factor diagrams.…”

“…The 2-norm has a close linear relationship with SRMSE due to values of SRMSE being 5 to 10 times those of the other two objective functions, and it does not have such a relationship with the other two objectives. The scattering of objectives and factors makes it difficult in decision making to select a single solution, because there is no clear trade-off solution (Blasco et al, 2008). However, compared with SOO, the Pareto solutions from MOO can make decision making easy, as it can be converted with expert opinion or some utility function.…”

Modeling and Multi-Objective Design Optimization of Quasi-Continuous High Magnetic Field Systems

“…Thus L should be corrected by a correction factor K correct . When the magnet is wound with copper and glass fiber, K correct is set as 1.06 and the more precise value of L is calculated by Equation (16).…”

“…Thus the point with the lowest norm may not be the preferred solution. To obtain a more preferred one, the graphical representation, combined with a coloring methodology of the points based on preferences of the designer, is adopted [16]. The Pareto points have different values for each objective that can be divided into several ranges according to the classification in Table 4.…”

A novel multi-objective optimization method for the pressurized reservoir in hydraulic robotics

“…Other studies built designer preference interactively by query to the designer (Pedro and Takahashi, 2013) or comparison of pairwise solutions by the designer (Branke et al, 2015;2016); in these schemes the designer is provided with only fractional information, instead of a big picture of the optimization potential in the current situation. The visualization of Pareto-optimal solutions is also often studied in specialized topics so that the designer can make decisions based on that visual information (Kollat and Reed, 2007;Blasco et al, 2008). Current graphical studies are mostly focused on objective space display of the Pareto front, and the relationship between the objective and the solution distribution is omitted.…”