A beginner’s guide to narrating the multiobjective tradeoff story

A posteriori decision making aided by state of the art multiobjective evolutionary algorithms can improve upon traditional a priori weighting techniques for solving complex engineering problems. Though advanced computing power is essential for enumerating the Pareto front of “non-dominated” solutions, computers are not able to perfectly capture stakeholder preferences or choose solutions that best fit those preferences. In a multiobjective decision setting, it is the job of the analyst to guide decision makers through the multiobjective search process and allow them to understand the logical progression of the analysis. In other words, it is the job of the analyst to use the MOEA results to narrate the “multiobjective tradeoff story”. Visual analytics are a key tool in this narration, particularly the use of parallel axis and 3-D scatter plots. This post will introduce some basic visual analytic tools and explain some helpful techniques that may be used for this narration. Examples are provided from 200 Pareto optimal solutions generated by MOEA search on a 5 objective problem formulation, where each objective was to be minimized.

Visualizing tradeoffs: 3-D scatter plots and parallel axis plots

Heterogeneous plots have been found to improve user’s ability to comprehend complex data sets. Two types of plots that can be used in tandem are parallel axis plots and 3D-scatter plots. 3D-scatter plots allow decision makers to plot 3 objectives against each other in 3 dimensional plot. The 3D-scatter plot is not limited to 3 objectives however; analysts can include additional objectives by modifying the size, color, shape and orientation of the plotted points. A 3D-scatter plot of the example data generated using Matlab can be found below.


In the figure above, objective 1 is plotted on the x-axis, objective 2 on the y-axis,  and objective 3 on the z-axis. Objective 4 is displayed through point size and objective 5 is displayed through point color

Another helpful visualization tool is the parallel axis plot. Parallel axis plots display each objective on a separate axis, whose scale is normalized to the range of scales given in the objective values. Each of solution is plotted as a line across the axes, with the objective function value for each objective plotted on its respective axis. Crossing lines represent tradeoffs between objectives for two different solutions. Two key benefits of parallel axis plots are their ability to scale to an unlimited number of dimensions and present different coordinates in a uniform manner. A parallel axis plot for the given data can be found below.

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The above plot shows the 5 objectives on parallel axes. Each line represents a different solution from the Pareto set. Crossing lines indicate tradeoffs between objectives. Line colors are based off the objective 1 value for each solution. The axis furthest to the right represents the row number for each solution in the csv file containing solutions.

The above plot was generated using the Web Cornell Tool, which can be found here: http://reed.cee.cornell.edu/parallel-axis/. This simple and easy to use tool allows decision makers to quickly make visually appealing parallel axis plots without writing a single line of code. To create a plot, you only need a csv file of Pareto optimal solutions. A detailed description of the tool can be found in Bernardo’s original post: https://waterprogramming.wordpress.com/2015/03/24/creating-parallel-axes-plots/

Narrating the MO Tradeoff Story Chapter 1: Plotting the baseline

Once you’ve learned how to create your 3-D scatter and parallel axis plots, you are ready to begin narrating the multiobjective tradeoff story. A key advantage of using a multiobjective problem formulation is that they may bring to light tradeoffs inherent to the system that decision makers had not previously been aware of. Decision makers are often hesitant to deviate from their current or “baseline” course of action without substantial evidence that alternate courses of action may be beneficial. By evaluating the baseline course of action along with alternate solutions from a search based on the multiobjective formulation, analysts can explicitly examine the tradeoffs they currently face, and objectively judge them against those discovered through search. The 3-D scatter plot and parallel axis plots below show how this may be done.

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In this plot, the baseline solution is shown as the diamond. 

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The baseline solution is highlighted in red, while the solutions found through MOEA search are plotted in the background.

Through use of the two plots above, the analyst can make a compelling argument to decision makers that that deviating from the baseline solution may be of beneficial. It is apparent that the baseline does very well in objectives 1, but rather poorly compared to the solutions found through MOEA search in objectives 2, 3, 4 and 5. Scenarios like this may result from baseline operating procedures that were based on single objective problem formulations where between conflicting objectives were not fully understood.

Chapter 2: Narrowing the field of potential solutions

Once decision makers have been made aware of the tradeoffs that are inherent to the baseline solution, the analyst can begin to guide them to find the most preferable solution from the Pareto optimal set. One potential pitfall of discovering solutions through MOEA search is the sheer number of Pareto optimal solutions that may be generated. A helpful tool to narrow the field of potential solutions is the technique known as brushing. To “brush” the data, decision makers simply choose acceptable/desirable ranges of each objective value and then filter the data to display only the Pareto optimal values that fall within that range. The Web Cornell Tool has two simple to use features that make it easy for the user to brush the data. Users can simply select acceptable values on each objectives axis or create 2-D vectors between objectives that specify the slope of tradeoff lines. A brushed version of the parallel axis plot and scatter plots can be found below. The brakes on the parallel axes depict the ranges of solutions deemed “acceptable” for consideration.

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A brushed version of the parallel axis plot leave decision makers with a much more manageable set of potential solutions


A brushed version of the 3D-scatter plot.

Chapter 3 and beyond…

The narration of the multiobjective tradeoff story is not meant to be a method for the analyst to dictate which decision should be made, but rather to provide insight into the nature of tradeoffs within those decisions. It is then up to the decision makers to use this information to come up with a policy that most closely fits their preferences. This post has provided two simple ways of using visual analytics to understand multiobjective tradeoffs and two techniques for narrating the “tradeoff story”. Future posts will take a deeper dive into multiobjective tradeoff analysis, including the search for system robustness and methodology for mapping the effects of solutions to different stakeholders.

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