Palaniappan Ramu

Visualization techniques in design space exploration with high dimensional data are helpful in enhancing the decision making in the context of multiple objective optimization. Visualization of Pareto solutions obtained is crucial to understand the trade-off between the objectives as it enables intuitive decision making. However, such a task is not trivial beyond three dimensions. In this work, we propose using interpretable self-organizing map (iSOM), to visualize Pareto solutions for MOO problems involving n objectives (n> 3 ). iSOM enable simplified component plane plots that allow visual inspection of the Pareto fronts and also allow identifying clusters in the Pareto front and the corresponding design variables. Proposed approach is successfully demonstrated on 3 analytical examples.

The goal of a multi-criteria decision-making (MCDM) approach is to select one or a few preferred solutions in an iterative manner from a set of Pareto-optimal solutions obtained by a generative or simultaneous evolutionary multi-and many-objective optimization (EMO and EMaO) algorithm. In each iteration, the decision-maker (DM) formulates a suitable scalarized optimization problem using preference information that guides the DM to arrive at the most desirable solution set. Visualization of trade-offs among multiple objectives and their interactions with constraints can provide crucial decision-making information. In this paper, we propose a visualization-assisted MCDM approach that utilizes interpretable Self-Organizing Maps (iSOM) on a well-known MCDM technique known as Pareto Race. The proposed method, applied to one test and two real-world problems involving three to five objectives, demonstrates the usefulness of the iSOM-visualization method in implementing Pareto Race decision-making approach. The study opens up further avenues for integrating iSOM visualization approach with other MCDM techniques.

In multi-criterion optimization, decision-makers (DMs) are not often interested in the complete Pareto-optimal front. Instead, they have preferences favoring specific parts of the front. Multi-criterion decision-making (MCDM) literature provides a plethora of approaches for introducing DM's preference information in an interactive manner to solve multi-criterion optimization problems. Interactions with DMs can be aided with a user-friendly visualization method or by using special data analysis procedures. An earlier study has indicated the use of self-organizing maps (SOM) as a tool for analyzing Pareto-optimal solutions. In this paper, we demonstrate how a specific MCDM method – NIMBUS – can be executed with the interpretable SOM (iSOM) approach iteratively to arrive at one or more preferred solutions. A visual illustration of the entire high-dimensional search space into multiple reduced two-dimensional spaces allows DMs to have a better understanding of the interactions of the objectives and constraints independently, and execute the NIMBUS decision-making procedure with a more wholistic approach. The paper demonstrates the proposed method on a number of multi- and many-objective numerical and engineering problems. The approach is now ready to be integrated with other popular MCDM methods.

Evolutionary multi-objective optimization (EMO) algorithms are predominantly used for solving multi- and many-objective optimization problems to arrive at the respective Pareto front. From a practical point of view, it is desirable for a decision-maker (DM) to consider objective vectors that are less sensitive to the small perturbation in design variables and problem parameters. Such insensitive, yet closer to Pareto-optimal solutions, lie on the so-called robust front. In real-world applications, such as engineering design and process optimization problems, perturbations in variables come from manufacturing tolerances, uncertainties in material properties, variations in operating conditions, etc. The existing EMO literature on robustness studies emphasized on finding the entire robust front, but hardly considered robustness in both optimization and decision-making tasks. In this paper, we propose and evaluate different algorithmic implementations of three aspects-multi-objective optimization, robustness consideration, and multi-criterion decision-making-together. Based on experimental results on two to eight-objective problems, we discuss the outcomes and advantages of different integration approaches of these three aspects and present the most effective combined approach. The results are interesting and should pave the way to develop more efficient multi-objective robust optimization and decision-making (MORODM) procedures for handling practical problems with uncertainties.

Visualizing and analyzing multiple Pareto-optimal solutions obtained using an evolutionary multi- or many-objective optimization algorithm is as important a task as the task of finding them. Besides helping to choose a single preferred solution, they provide a better understanding of the trade-off among the objectives and also reveal key insights about interactions among variables and objectives for the Pareto-optimal solutions. Existing visualization methods do not provide a comprehensive account of both visualization and analysis of Pareto-optimal solutions. In this paper, we present an interpretable self-organizing map (iSOM) method that produces a more simplistic mapping of higher-dimensional variable spaces into two dimensions. Multiple iSOM plots, one for each objective, allows an easier visual understanding of trade-off among objectives. By identifying high trade-off Pareto-optimal solutions and marking them on the iSOM plots, we also provide decision-makers a comprehensive method to locate critical and likely preferred solutions on the Pareto-optimal front. The visualization and analysis of Pareto-optimal solutions using iSOMs are demonstrated on 11 problems involving three to five objectives. As discussed, the approach is generically applicable to higher-dimensional and constrained problems.