IEEE Computational Intelligence Magazine - May 2020 - 46

The PaletteViz plot allows a DM to look at only about
20 critical points (marked in red color) from a set of 1,292
non-dominated points. If the DM prefers boundary points
(for which at least one of the objectives has an extreme
value) a few red-circled points can be further looked into,
however if the DM prefers a point which lies at the core of
the data-set (with a good compromise of all four objectives)
then three red-circled points in Layer 3 (from top) can be
further looked into.
D. Disjointed Pareto-Optimal Front

As we have discussed before, in the presence of a Pareto set having multiple disjointed clusters in the original high-dimensional
objective space, a DM may be interested in the boundary points
of each cluster.To illustrate this scenario, we have used the palette

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visualization on three- and five-objective C2DTLZ2 problem
[26]. The Pareto set consists of multiple clusters that are generated by applying (m + 1) constraints on DTLZ2 [30] problem. A
three-dimensional scatter plot of C2DTLZ2 is presented in Figure 13. Three extreme points have a large trade-off, due to the
concavity of the Pareto-optimal surface. Figure 14 shows the
PaletteViz plot for a five-objective C2DTLZ2 problem. Six
clusters and their relative locations are noticeable from the PaletteViz plot. The pink points are closer to a constraint boundary
and cyan colored points are well inside the feasible region. It is
interesting to note that the top-most layer has mixed pink and
cyan-colored, providing vital information to a DM. Not all
boundary points are on constraint surfaces. If a boundary point
near the constraint surface is desired, the DM can only analyze
the pink points further. Otherwise, if a core point near the constraint surface is desired, a pink point from one of the bottom
layers can be chosen.
In order to compare PaletteViz plot with another existing
visualization method, we plot the same data-set for the fiveobjective C2DTLZ2 problem in a heatmap [9] visualization in
Figure 15. For this particular application, heatmap does not
provide much information about six clusters that the data possess and nearness of any point to a constraint boundary.

VI. PaletteViz for Engineering Design Problems
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So far, we have seen the proposed palette visualization technique can address different scenarios that a DM may come
across during a decision-making process. Next, we apply PaletteViz on two engineering design problems.

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A. Vehicle Crash-Worthiness Problem

FIGURE 13 Pareto set for the three-objective C2DTLZ2 problem.
There are four isolated regions on the data-set. The data-set contains
2,373 points.

We apply our method to the vehicle crash-worthiness problem
from [18].This is a three-objective optimization problem, however the Pareto set in this problem is observed to have two separate

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FIGURE 14 PaletteViz plot of the five-objective C2DTLZ2 problem
indicating five extreme and one central cluster in the original data-set
containing 5,170 points.

IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | MAY 2020

FIGURE 15 The heatmap [9] visualization of five objective C2DTLZ2
problem. This visualization does not convey any interesting information about the Pareto set. All the points are sorted in descending
order with respect to f1. The cluster closer to extreme f1 value has a
large difference in f1 values from other clusters.

IEEE Computational Intelligence Magazine - May 2020

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