IEEE Computational Intelligence Magazine - May 2020 - 39

u1 =

f j cos (a j)

j=1
m

, u2 =

j=1
m

❏ Active points lying on constraint boundaries or any other

f j sin (a j)
fj

j=1

problem-specific desired points

(1)

III. Visualization of a Pareto Set From a Decision
Maker's Perspective

Most existing visualization methods used in EMO studies
today are borrowed from the general high-dimensional data
visualization literature. From our previous experience with
industry collaborations, we have observed that a decision
making procedure requires a few unique functional considerations which the usual high-dimensional data visualization
methods may not have considered to be important. Thus, it is
time that EMO researchers develop new and more useful
visualization techniques for visualizing the trade-off within
data points in a more spacially interpretable way, rather than
simply following existing methods from the literature. Therefore, the next important question is what are these unique
functionalities that a multi-objective data visualization technique must have, so that only a few meaningful and relevant
Pareto-optimal points can be presented to DMs for a faster
and worthy decision analysis.
So far, the EMO literature has shown lukewarm interest in
devising efficient methods for choosing a single preferred solution. This is probably due to the subjective, often nonanalytic,
considerations associated with the task. We describe a few such
scenarios here:
❏ Points with large trade-off and knee points
❏ Boundary points
❏ Isolated points

1) Points with Large Trade-off and Knee Points
Perhaps, the most desirable aspect of Pareto-optimal points is
the points in the objective function space that offer most tradeoff among all other points in the entire set [20], [21]. The
trade-off of a point can be defined in many ways, but it
involves the location of the point with respect to its neighbors
in the objective space. A point having a large trade-off means
the gain in moving to a neighbor is small compared to the loss
in other objective values. Thus, there is not much motivation
to make the move, as loss outweighs gain, and the point having
a large trade-off value is most desired to the decision-makers.
This is fairly commonsensical in most decision-making tasks
and it would be revealing to the DMs if such trade-off information is directly provided through a visualization technique.
Towards this goal, we suggest the following. The trade-off of a
Pareto point can be computed in two steps: (i) identify neighbors, and (ii) compute the trade-off. Since the nature of objectives and their relative values are well understood by DMs, a
minimum threshold on the trade-off value can be set by DMs
and the Pareto-optimal points which exceed the threshold can
be presented clearly in a visualization technique so that DMs
can easily recognize these special data points from hundreds of
other Pareto-optimal points for further considerations.
In this paper, we follow the definition of knee points (i.e.,
points with comparatively larger trade-off) discussed in [20]. Let
us consider the simple Pareto set depicted in Figure 2, with
three objectives which are to be minimized. The data-set has a
clearly visible bulge in the middle. If we assume linear preference functions, and furthermore assume that each preference

75
Centroid of
Cluster B

0.225

25
Cluster B

0.175

-25

0.125

Cluster A

0.075
0.025

1,660
1,670
7

8
f2

9
(a)

1,680 f1
1,690

-50
-75
-100
-80

Centroid of
Cluster B
-60

-40

-20

0
u1

(b)

FIGURE 1 (a): Three-dimensional Pareto set for the vehicle crash-worthiness problem [18]. There are two separate clusters in the data-set. The
light green points are the Pareto-optimal points that are close to the cluster centroids and dark blue points are close to the cluster boundaries.
(b): t-SNE mapping of the same data-set. Here, we see that the neighborhood relationships are completely distorted in cluster B. The t-SNE mapping divides cluster B into two different clusters, resulting in a total of three clusters in the two-dimensional mapped space. Points that are close
to the cluster B centroid are separated by some boundary points. Also, one boundary point of cluster B with high trade-off (red color) is placed
close at the farthest corner of cluster A. There are 4,450 data points in this data-set.

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