How to Read Many-Objective Solution Sets in
Parallel Coordinates
Miqing Li, Liangli Zhen, and Xin Yao1

arXiv:1705.00368v1 [] 30 Apr 2017

1 CERCIA,

School of Computer Science, University of Birmingham, Birmingham B15 2TT, U. K.
∗ Email: limitsing@gmail.com, llzhen@outlook.com, x.yao@cs.bham.ac.uk

Abstract: Rapid development of evolutionary algorithms in handling many-objective optimization problems requires viable methods of visualizing a high-dimensional solution set. Parallel coordinates which
scale well to high-dimensional data are such a method, and have been frequently used in evolutionary
many-objective optimization. However, the parallel coordinates plot is not as straightforward as the classic
scatter plot to present the information contained in a solution set. In this paper, we make some observations of the parallel coordinates plot, in terms of comparing the quality of solution sets, understanding the
shape and distribution of a solution set, and reflecting the relation between objectives. We hope that these
observations could provide some guidelines as to the proper use of parallel coordinates in evolutionary
many-objective optimization.
Keywords: Many-objective optimization, visualization, parallel coordinates

1 Introduction
The classic scatter plot is a basic tool in viewing solution vectors in multiobjective optimization. It
allows us to observe/perceive the quality of a set of solutions, the shape and distribution of a set of
solutions, the relation between objectives (e.g., the extent of their conflict), etc. Unfortunately, the
scatter plot may only be done readily in a 2D or 3D Cartesian coordinate space1 . It could be difficult
for people to comprehend the scatter plot in a higher-dimensional space.
An alternative to view data with four or more dimensions is using parallel coordinates [2–4] (also
called value paths [5]). Parallel coordinates display multidimensional data (a set of vectors) in a twodimensional graph, with each dimension of the original data being translated onto a vertical axis in
the graph, and a vector is represented as a polyline with vertices on the axes. As a visualization tool,
parallel coordinates have received modest attention in the early stage of evolutionary multiobjective
optimization (EMO) [6, 7]. As many-objective optimization (i.e., the number of objectives to be
optimized being larger than three [8, 9]) becomes a new research topic in the EMO area, there
has been increasing interest in presenting solution vectors in the high-dimensional space. Parallel
coordinates which are scalable to objective dimensionality naturally become a good alternative to do
so [10]. Now the parallel coordinates plot has been dominantly used in many-objective optimization
despite the emergence of various visualization techniques [5, 11–13]. This includes it being used to
investigate the search behavior of algorithms [14–16], to examine preference-based search [17–19], to
compare different solution sets [20–22], to verify performance metrics [23–25], and furthermore to
help design new many-objective optimizers [26, 27].
1

The scatter plot matrix [1] is scalable to any dimension, but it only reflects the relation between two objectives.

1

50

𝒂

𝒃

𝒄

40
30
20
10
𝟏

𝟐
𝟑
Dimension No.

𝟒

Figure 1: The parallel coordinates plot of three four-dimensional points a (15, 31, 20, 50),
b (10, 18, 2, 30) and c (20, 5, 32, 20).
Despite the popularity, the parallel coordinates plot is not as straightforward as the scatter plot
in presenting the information contained in a solution set. Due to mapping multi-dimensional data
onto a lower 2D space, the loss of information is inevitable. This could naturally lead to questions
raised. Specifically, in the context of multiobjective optimization, one may ask
• Can the parallel coordinates plot indicate the quality of a solution set, e.g., its convergence,
extensity, uniformity and coverage?
• Can the parallel coordinates plot imply the shape and distribution of a solution set? In other
words, what can we see from the pattern of solution lines in parallel coordinates?
• How much information can the parallel coordinates plot tell in terms of the relation among
objectives? To be specific, does the order of objectives displayed in parallel coordinates matter?
In this paper, we make some observations on the above questions, attempting to provide some
guidelines as to the use of parallel coordinates in evolutionary multiobjective optimization. The rest
of the paper is organized as follows. Section II briefly introduces parallel coordinates. Sections III–V
are devoted to answering those three questions, respectively. Section VI describe how to draw a
parallel coordinates plot. Section VI concludes the paper and presents some possible future research
lines.

2 Parallel Coordinates
To show a set of points in an m-dimensional space, parallel coordinates map them onto a 2D graph,
with m parallel axes being plotted, typically vertical and equally spaced. A point in an m-dimensional
space is represented as a polyline with vertices on these parallel axes, and the position of the vertex
on the i-th axis corresponds to the value of the point on the i-th dimension. Parallel coordinates are
simple to construct and scale well with the dimensionality of data. Adding more dimensions only
involves adding more axes. Figure 1 presents an example of the parallel coordinates plot, where three
4D points are mapped to three polylines, respectively.

3 Quality Measuring
Given dimensionality reduction in the mapping of parallel coordinates, some loss of information
is expected. In this section, we will see what and how much information parallel coordinates can
preserve and reflect in terms of the quality of a solution set in multiobjective optimization.

2

3 .5

3 .0

3 .0

O b je c tiv e V a lu e

O b je c tiv e V a lu e

3 .5

2 .5
2 .0
1 .5
1 .0
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1

2

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1 0

O b je c tiv e N o .

1

2

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5

6

7

8

9

1 0

O b je c tiv e N o .

(a) NSGA-II (GD+ =2.26E-1)

(b) GrEA (GD+ =1.20E-2)

Figure 2: The solution set obtained by NSGA-II and GrEA on the 10-objective DTLZ2, and their
evaluation results on the convergence measure GD+ (the smaller the better).
Often, the quality of a solution set in multiobjective optimization can be reflected via four measures:
convergence, coverage, uniformity, and extensity. Convergence of a solution set measures the closeness
of the set to the Pareto front; coverage considers the region of the set covering in comparison with
the whole Pareto front; uniformity quantifies the distance between neighboring points in the set;
and extensity refers to the range of the set. In general, there is no clear conceptual difference of
these quality measures between many-objective optimization and multi-objective optimization with
two or three objectives. However, many-objective optimization typically poses bigger challenge for
evolutionary algorithms to achieve a good balance among these aspects.
A straightforward feature that parallel coordinates can tell is the the extensity of a solution
set. This feature can make parallel coordinates replace extensity metrics in EMO, e.g., maximum
spread [28]. In the following, we will discuss if parallel coordinates can reflect other aspects of a
solution set’s quality, i.e., the convergence, coverage and uniformity.

3.1 Convergence
In multiobjective optimization, Pareto dominance is a fundamental criterion to compare solutions in
terms of convergence. Parallel coordinates can clearly reflect the Pareto dominance relation between
two solutions (such as polyline a being dominated by polyline b in Figure 1, assuming a minimization
problem scenario) if the solution polylines are not overcrowded. It is worth mentioning that one can
remove dominated solutions in parallel coordinates if they are only interested in nondominated ones.
This may make the plot clearer especially when comparing the quality of solution sets.
In addition to reflecting the Pareto dominance relation, the range that a solution set in parallel
coordinates can largely imply its convergence. Figure 2 is such an example, where the parallel
coordinates representation of two solution sets obtained by one run2 of two EMO algorithms, NSGAII [29] and GrEA [30], on the 10-objective DTLZ2 problem [31] is shown. As can be seen, NSGA-II
fails to converge, with its solution set ranging from 0 to around 3.5 in contrast to the problem’s
Pareto front ranging from 0 to 1. GrEA has a good convergence on this problem and its solution set
has the same range as the Pareto front. These observations can be confirmed by the results of the
convergence metric GD+ [32] shown in the figure. GD+ is a modified version of the original GD [33],
which makes it compatible with Pareto dominance.
However, we may not be able to accurately know the convergence of solution sets by their range
shown in parallel coordinates. That is, even if two solution sets are located in the same range, they
can perform considerably differently in terms of convergence. Figure 3 gives such an example, where
2

The setting of the population size and maximum evaluations was 100 and 30,000, respectively. This setting was used
in all conducted experiments in this paper, unless explicitly mentioned otherwise. In addition, the grid division in
GrEA was set to 8.

3

2 0

1 5

1 5

O b je c tiv e V a lu e

O b je c tiv e V a lu e

2 0

1 0

5

0
1

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9

1 0

5

0

1 0

1

2

3

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O b je c tiv e N o .

5

6

7

8

9

1 0

O b je c tiv e N o .

(a) NSGA-II (GD+ =3.63E-1)

(b) GrEA (GD+ =6.32E-2)

1 .0

1 .0

0 .8

0 .8

O b je c tiv e V a lu e

O b je c tiv e V a lu e

Figure 3: The solution set obtained by NSGA-II and GrEA on the 10-objective WFG7, and their
evaluation results on the convergence metric GD+ (the smaller the better).

0 .6

0 .4

0 .2

0 .0
1

2

3

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6

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O b je c tiv e N o .

1

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4

5

6

7

8

9

1 0

O b je c tiv e N o .

(a) AR

(b) IBEA

Figure 4: The solution set obtained by AR and IBEA on the 10-objective DTLZ2.
solution sets obtained by one run of NSGA-II and GrEA on the 10-objective WFG7 problem [34]
are shown. As seen, both algorithms virtually reach the range of the Pareto front (from 0 to 2i
where i is the objective index of the problem), but they have different GD+ results. NSGA-II is
returned a significantly higher (worse) GD+ value than GrEA. This occurrence can be caused by two
possibilities. One is that the solution set of NSGA-II is not actually close to the Pareto front. The
other is that most of solutions in the set converge already while a small portion of set is far away
(but still in the range of the Pareto front).

3.2 Coverage
In parallel coordinates, it is straightforward to see which region a solution set does not reach on any
objective3 . For example, in Figure 4 the solution set obtained by the AR method [35] concentrates
in one tiny area and the set by IBEA [36] fails to cover the first six objectives on the 10-objective
DTLZ2. Moreover, we can conjecture some distribution features of solution sets from their parallel
coordinates representation. Take the solution sets of Figure 5 as an example; their parallel coordinates representation is shown in Figure 6. From Figure 6, we can know that the solution sets of
IBEA and SMS-EMOA [37] fail to cover the region between 0 and 0.2 on all three objectives. Also,
most of the solutions obtained by IBEA are located in the boundary of the Pareto front as there are
very few lines distributed around the middle section on all three objectives in the figure.
However, there do exist some cases that different solution sets have the same parallel coordinates
plots. We can easily construct such an example. In Figure 7, solution set B has a better coverage
3

Note that for real-world problems whose Pareto front is unknown, we cannot tell if a solution set reaches the optimal
region of objectives or not.

4

(a) IBEA

(b) SMS-EMOA

(c) MOEA/D

1 .0

1 .0

0 .8

0 .8

0 .8

0 .6
0 .4
0 .2
0 .0
1

2

3

O b je c tiv e V a lu e

1 .0

O b je c tiv e V a lu e

O b je c tiv e V a lu e

Figure 5: The solution set obtained by IBEA, SMS-EMOA and MOEA/D on the 3-objective DTLZ2,
shown in Cartesian coordinates.

0 .6
0 .4
0 .2
0 .0

O b je c tiv e N o .

1

2

3

0 .6
0 .4
0 .2
0 .0
1

O b je c tiv e N o .

(a) IBEA

2

3

O b je c tiv e N o .

(b) SMS-EMOA

(c) MOEA/D

Figure 6: The corresponding parallel coordinates of the solution sets in Figure 5.

(a) Solution set A

(b) Solution set B

Figure 7: An artificial example of two solution sets (A and B) having same parallel coordinates plots
shown in Figure 8.

5

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O b je c tiv e V a lu e

2 .0

1 .0

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0 .0
1

2

3

1

2

O b je c tiv e N o .

3

O b je c tiv e N o .

(a) Solution set A

(b) Solution set B

2 0

2 0

1 5

1 5

O b je c tiv e V a lu e

O b je c tiv e V a lu e

Figure 8: The parallel coordinates of the solution sets in Figure 7.

1 0

0

5

1

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9

1 0

0

5

1 0

O b je c tiv e N o .

1

2

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9

1 0

O b je c tiv e N o .

(a) NSGA-II (DCI=7.17E-1)

(b) GrEA (DCI=7.75E-1)

Figure 9: The solution set obtained by NSGA-II and GrEA on the 10-objective WFG7, and their
evaluation results on the coverage metric DCI (the bigger the better).
than set A (the four solutions in set A being duplicate), but the two sets have the same pattern in
parallel coordinates (Figure 8). Note that if we change the order of some objectives (e.g., f1 and f2 ),
the parallel coordinates of the two solution sets in this example would be different.
One important fact that we would like to note is that as parallel coordinates map an m-dimensional
graph onto a 2D graph they cannot fully reflect the coverage of solution sets. A set of solutions
(represented by polylines) may have a good coverage over the range of the Pareto front in the 2D
graph, but they may only cover part of the Pareto front in the original m-dimensional space. An
interesting example is shown in Figure 9. In that figure, NSGA-II appears to have a better coverage
than GrEA according to the parallel coordinates plots, but GrEA has a better coverage evaluation
result, measured by the coverage metric Diversity Comparison Indicator (DCI) [38].

3.3 Uniformity
In parallel coordinates, it is not easy to see how evenly a set of solutions are distributed. However,
a set of uniformly-distributed polylines in parallel coordinates often implies a uniformly-distributed
solution set. As shown in Figures 10 and 11, MOEA/D [39] has a perfectly-distributed solution
set and its corresponding polylines in parallel coordinates are distributed uniformly and regularly.
This is in contrast to the solution set of NSGA-II which are distributed rather irregularly in both
Cartesian and parallel coordinates plots. Note that a set of irregularly-distributed polylines may
not represent a badly-distributed solution set, as uniformly-distributed solutions can have distinct
values on different objectives. To show this, we select two EMO algorithms, MOEA/D [39] and
BCE-MOEA/D [40], both of which are able to obtain a uniformly-distributed solution set on DTLZ2
(see Figure 12). In MOEA/D, the population distribution is maintained by a set of systematically-

6

(a) NSGA-II

(b) MOEA/D

0 .5

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O b je c tiv e V a lu e

O b je c tiv e V a lu e

Figure 10: The solution set obtained by NSGA-II and MOEA/D on the 3-objective DTLZ1, shown
in Cartesian coordinates.

0 .3
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1

2

0 .3
0 .2
0 .1
0 .0

3

O b je c tiv e N o .

1

2

3

O b je c tiv e N o .

(a) NSGA-II

(b) MOEA/D

Figure 11: The corresponding parallel coordinates of the solution sets in Figure 10.
generated, uniformly-distributed weight vectors (within a simplex), and thus ideally its solutions only
take several equivalent values on all the objectives. In contrast, in BCE-MOEA/D the population
distribution is maintained by a niching-based criterion, and thus its solutions can spread over the
whole range for each objective. Figure 13 gives the solution sets obtained by MOEA/D and BCEMOEA/D on the 10-objective DTLZ24 . As seen, on the uniformity metric Spacing (SP)5 [41],
BCE-MOEA/D even performs better than MOEA/D, but we cannot see this from their parallel
coordinates representation in the figure. This phenomenon may happen frequently when comparing
decomposition-based algorithms having a set of systematically-generated weight vectors (such as
MOEA/D and NSGA-III [42]) with algorithms that do not use such decomposition techniques (such
as SPEA2 [43] and Two arch2 [44]). So care needs to be taken when making a conclusion about the
distribution uniformity of solution sets from parallel coordinates.
Finally, it is worth mentioning that parallel coordinates plots can be easily cluttered with multiple lines overlaid. This may completely prevent solution sets’ distribution from being observed.
Figures 14 and 15 show such an example, with two solution sets obtained by NSGA-II and SPEA2
on the 10-objective ML-DMP problem [45, 46]. The m-objective ML-DMP minimizes the distance
4

Here the number of DTLZ2’s decision variables is set to m − 1 (m is the number of objectives) to ensure that
all solutions produced by algorithms are Pareto optimal; thus the uniformity measure cannot be affected by the
difference of solution sets’ convergence.
5
In this paper, the SP metric has been slightly modified to make it compatible with Pareto dominance. That is, if
two solution sets are comparable in terms of Pareto dominance, then the SP value of the dominating set is 0 and
the SP value of the dominated set is 1.

7

(a) MOEA/D

(b) BCE-MOEA/D

1 .0

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0 .8

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O b je c tiv e V a lu e

O b je c tiv e V a lu e

Figure 12: The solution set obtained by MOEA/D and BCE-MOEA/D on the 3-objective DTLZ2,
shown in Cartesian coordinates.

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O b je c tiv e N o .
(a) MOEA/D (SP=1.05E-1)

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1 0

O b je c tiv e N o .
(b) BCE-MOEA/D (SP=7.74E-2)

Figure 13: The solution set obtained by MOEA/D and BCE-MOEA/D on the 3-objective DTLZ2,
and their evaluation results on the uniformity metric SP (the smaller the better).
of two-dimensional points to a set of m straight lines, each of which passes through one edge of a
given regular polygon with m vertexes. One interesting characteristic of ML-DMP is that the points
in the regular polygon and their objective images are similar in the sense of Euclidean geometry. In
other words, the ratio of the distance between any two points in the polygon to the distance between
their corresponding objective vectors is a constant. This allows a straightforward understanding of
the distribution of the objective vector set via observing the solution set in the 2D decision space.
As can be seen in Figure 14, SPEA2 has a far better distribution uniformity than NSGA-II, but we
cannot see the difference between their parallel coordinates representation in Figure 15.

4 Solution Set Distributions
In parallel coordinates, it is straightforward to know the conflict between objectives. The number
of intersection lines between adjacent objectives reflects their conflicting degree. If there is no intersection of any pair of lines between adjacent objectives, then these two objectives are completely
non-conflicting (i.e., harmonious [47]), such as objectives f1 versus f2 and objectives f2 versus f3 in
Figure 16. If there are many lines intersecting, then the two objectives are heavily conflicting, such
as objectives f3 versus f4 and objectives f4 versus f5 in Figure 16. If any pair of lines intersect, then
the two objectives are completely conflicting to each other.
An interesting phenomenon in the parallel coordinates plot is that if all lines between two adja-

8

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(a) NSGA-II

(b) SPEA2

2 .0

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O b je c tiv e V a lu e

O b je c tiv e V a lu e

Figure 14: The solution sets (in the decision space) obtained by NSGA-II and SPEA2 on the 10objective ML-DMP where the search space is precisely the optimal polygon.

1 .0

0 .5

0 .0
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4

7

1 0

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O b je c tiv e N o .

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2

5

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O b je c tiv e N o .

(a) NSGA-II

(b) SPEA2

Figure 15: Parallel coordinates of the solution sets (in the objective space) in Figure 14.
cent objectives intersect at one point, then these two objectives are negatively linearly dependent.
Figure 17 is such an example. The four-objective ML-DMP problem minimizes the distance of points
to four lines passing through the four edges of the given rectangle. From this definition, we can see
that the two pairs of objectives, f1 versus f3 and f2 versus f4 , are negatively linearly dependent for
the solutions in the rectangle (f1 = −f3 , f2 = −f4 ). Therefore, each of the objective pairs intersects
at one point, as shown in Figure 17(b).
This property is the known duality between the parallel coordinates representation and the Cartesian coordinate representation of data [2,3]: points in Cartesian coordinates map into lines in parallel
coordinates, while lines in Cartesian coordinates map into points in parallel coordinates. Consider a
line ` : f2 = kf1 + b in the Cartesian coordinate plane of f1 and f2 , and consider two points lying on
this line, say (x, kx+b) and (y, ky +b) (shown in Figure 18(a)). Figure 18(b) shows the corresponding
parallel coordinates representation of the two points. For simplicity, let the distance between the
vertical axes f1 and f2 be 1; then it is easy to know that the two lines intersect at a point given by
ρ : ((1 − k)−1 , b(1 − k)−1 ) in parallel coordinates. This point depends only on k and b, the parameters
of the original line in the Cartesian plane. This indicates that the parallel coordinates representation
of any point on ` passes through the point ρ.
From the coordinates of the point ρ, we can see the relation between the position of ρ and the
slope k of the line `. If k < 0, the intersection occurs between the two parallel coordinates axes.
Especially, when k = −1 the intersection is precisely midway, as in the example of Figure 17(b). If
1 < k or 0 < k < 1, then the intersection point is on the left side or right side of the two coordinate
axes, respectively. When k = ±∞ or k = 0, the intersection point is on the left axis or right axis,
respectively. Finally, when k = 1, the lines are parallel between the two axes in parallel coordinates.
The above properties can help us understand the relation between objectives. For example, from the
parallel coordinates plot in Figure 16 we know that f1 = f2 and f2 = kf3 + b, where k > 1 and b = 0.

9

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O b je c tiv e V a lu e

0 .8
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O b je c tiv e N o .

Figure 16: The solution set obtained by SPEA2+SDE [48] on the 5-objective DTLZ5(I, M ) [49],
where I = 3.

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1 .2

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O b je c tiv e N o .

X 1

(a) Decision space

(b) Objective space

Figure 17: The solution set of SPEA2+SDE on the 4-objective ML-DMP.

𝑓2

𝑓2

ℓ

𝑓1

ℓ

𝑓1

𝑓2

(1, 𝑦)(1, 𝑦)

(𝑦, 𝑘𝑦
(𝑦,+𝑘𝑦
𝑏)+ 𝑏)

𝑓2
(2, 𝑘𝑦
(2,+𝑘𝑦
𝑏)+ 𝑏)
𝝆 𝝆
(2, 𝑘𝑥
(2,+𝑘𝑥
𝑏)+ 𝑏)

(𝑥, 𝑘𝑥
(𝑥,+𝑘𝑥
𝑏)+ 𝑏)
(1, 𝑥)(1, 𝑥)

𝑓1

𝑓1

Objective
Objective
No. No.

(a) Cartesian coordinates

(b) Parallel coordinates

Figure 18: An example of a line ` : f2 = kf1 + b in the Cartesian coordinate plane corresponding to
a point ρ : ((1 − k)−1 , b(1 − k)−1 ) in the parallel coordinates plane.

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O b je c tiv e V a lu e

0 .4
0 .3
0 .2
0 .1
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2

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O b je c tiv e N o .

(a) Cartesian coordinates

(b) Parallel coordinates

Figure 19: The solution set obtained by MOEA/D on the 3-objective DTLZ1.
Finally, note that since the horizontal position of the intersection point in parallel coordinates
depends only on the slope k, when we can see many lines between two objectives in the parallel
coordinates plot intersecting at same horizontal position (but at different vertical positions), this
means that many lines connecting two points in the Cartesian coordinate space have the same slope
with respect to these two objectives. This occurs often when points in Cartesian coordinates are
absolutely uniformly-distributed on the plane of these two objectives. The solution set in Figure 19
has such a pattern (see the midway of two adjacent objectives in Figure 19(b)). More interesting
correspondence between the patterns of lines in parallel coordinates and the relation of objectives in
the solution set can be found in [4].

5 Objective Order in Parallel Coordinates
In parallel coordinates, each axis has at most two neighboring axes (one on the left and one on the
right). Different order of objective axes presents different information with respect to the relation
between objectives. Take Figure 20 as an example. In Figure 20(a) where the order of objectives
is f1 , f2 , f3 , f4 , f5 , the conflict between any two adjacent objectives is rather weak. In contrast, in
Figure 20(b) where the order of objectives is f1 , f3 , f5 , f2 , f4 , the conflict between any two adjacent
objectives is quite intense.
In a solution set with m objectives, its parallel coordinates representation can only
 show m − 1
relationships at a time. This can be a very small portion compared to the total m
2 relationships
existing in m objectives. Therefore, a good objective axis arrangement providing the user as much
(clear) information as possible is of importance. As shown in Figures 21 and 22, after swapping some
objectives, we can see interesting patterns (linearly dependent) between some pairs of objectives.
However, determining a good order of axes is nontrivial. Researchers in the data visualization area
mainly focus on clutter reduction techniques [50, 51]. In the multiobjective optimization domain, an
attempt has been made to place the most harmonious objectives in a row [52].

6 How to Draw A Parallel Coordinate Plot
In this section, we give procedures of how to plot a solution set in parallel coordinates by several
commonly-used graphing tools: MS Excel, MATLAB, Latex, and Origin. Tables 1–4 provide the
steps/codes by MS Excel, MATLAB, Latex, and Origin, respectively. For the command-line format
tools (i.e., MATLAB and Latex), we use the example in Figure 1. Figure 23 also presents the graphs

11

1 .8

1 .5

1 .5

O b je c tiv e V a lu e

O b je c tiv e V a lu e

1 .8

1 .2
0 .9
0 .6
0 .3
0 .0
1

2

3

4

1 .2
0 .9
0 .6
0 .3
0 .0

5

1

3

5

O b je c tiv e N o .

2

4

O b je c tiv e N o .

(a)

(b)

1 .0

1 .0

0 .8

0 .8

O b je c tiv e V a lu e

O b je c tiv e V a lu e

Figure 20: The solution set of SPEA2+SDE on the 5-objective ML-DMP, shown by different order
of objectives in parallel coordinates.

0 .6
0 .4
0 .2
0 .0

0 .6
0 .4
0 .2
0 .0

1

4

3

5

2

1

2

O b je c tiv e N o .

3

4

5

O b je c tiv e N o .

(a)

(b)

1 .4

1 .4

1 .2

1 .2

O b je c tiv e V a lu e

O b je c tiv e V a lu e

Figure 21: The solution set of SPEA2+SDE on the 5-objective DTLZ5(I,M) where I = 3, shown by
different order of objectives in parallel coordinates.

1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
1

2

3

1 .0
0 .8
0 .6
0 .4
0 .2
0 .0

4

O b je c tiv e N o .

1

3

2

4

O b je c tiv e N o .

(a)

(b)

Figure 22: The solution set of SPEA2+SDE on the 4-objective ML-DMP, shown by different order
of objectives in parallel coordinates.

12

Table 1: Steps of creating a parallel coordinates plot in MS Excel.
(1) Input the data as a table with each row as a
solution and select them.
(2) Click Insert -> Recommended Charts.
(3) On the Recommended Charts tab, scroll
through the list of charts that Excel recommends for your data, click Line chart -> OK.
(4) Use the Chart Elements, Chart Styles, and
Chart Filters buttons next to the upper-right
corner of the chart to add chart elements like
axis titles or data labels, customize the look of
your chart, or change the data shown in the
chart.

Table 2: Codes of creating a parallel coordinates plot in MATLAB.
X = [15 31 20 50; 10 18 2 30; 20 5 32 20];
groups = {’a’, ’b’, ’c’};
parallelcoords(X,’group’,groups);
xlabel(’Objective No.’);

drawn by the four tools to that example. Finally, we would like to note that in this paper all of the
parallel coordinates graphs of data examples were drawn by Origin.

7 Conclusions
Parallel coordinates have drawn increasing attention in many-objective optimization, but mapping
a many-objective solution set onto a 2D parallel coordinates plane may not be straightforward to
reveal the information contained in the set. This paper has made some observations on the use of
parallel coordinates to present a solution set in many-objective optimization. In particular,
• the parallel coordinates representation of a solution set can partly reflect its convergence, coverage and uniformity. This suggests that parallel coordinates can be an assistant tool (but not
entirely replacing quality metrics) in assessing a many-objective solution set.
• although the clarity can be affected by overlapping polygonal lines, parallel coordinates transform certain geometrical features of a many-objective solution set into easily seen 2D patterns.
• the order of objective axes matters in parallel coordinates. To better present the relationship
between objectives, it may need to be rearranged according to features of the solution set
considered.
Our subsequent study is towards overcoming/alleviating the difficulties of interpreting parallel coordinates plots presented in this paper. Particularly, how to arrange the order of objectives will be the
focus of our future work as it had presented its usefulness in the paper. In this regard, a straightforward thought is to place the most conflicting objectives or the most harmonious objectives together

13

Table 3: Codes of creating a parallel coordinates plot in Latex using the PGFPlots package.
\usepackage{pgfplots}
\begin{tikzpicture}
\begin{axis}[xlabel={Objective No.},symbolic
x coords={1, 2, 3, 4},xtick=data]
\addplot+[mark=none,draw=green,sharp plot]
plot coordinates {(1,15) (2,31) (3,20) (4,50)};
\addplot+[mark=none,draw=blue,sharp plot]
plot coordinates {(1,10) (2,18) (3,2) (4,30)};
\addplot+[mark=none,draw=red,sharp
plot]
plot coordinates {(1,20) (2,5) (3,32) (4,20)};
\end{axis}
\end{tikzpicture}

Table 4: Steps of creating a parallel coordinates plot in Origin.
(1) Create a table consisting of the first column
being X axis from 1 to m (where m is the number
of objectives) and the remaining columns being
Y axis with each column for a solution.
(2) Select the table and click the Line button at
the lower-left corner of the panel.
(3) Double click the Axis Labels and the polylines of parallel coordinates to customize the
look of the chart.

Coordinate Value

Objective Values

50

40

a

b

c

Tabs. ??-?? provide the examples and codes of plotting
40
Fig. 1 in MS Excel, MATLAB, and Latex, respectively.

30
20

30
TABLE I

C REATING
20 A PARALLEL COORDINATES PLOT IN MS-E XCEL .

10
0

I. I NTRODUCTION

1

2
3
Objective No.

(a) MS Excel

4

50

5 0

40

4 0

O b je c tiv e V a lu e s

60

50

Objective Values

60

30
20
10

(1) Input10the data as a table with each row as a solution
and select them.
(2) Click0Insert -> Recommended Charts.
1
2
3
(3) On the Recommended Charts tab, scroll through the
Objective
No. data, click
list of charts that Excel recommends
for your
line chart -> OK.
(4) Use the Chart Elements, Chart Styles, and Chart
Filters buttons next to the upper-right corner of the chart
to add chart elements like axis titles or data labels,
customize the look of your chart, or change the data
thats shown in the chart.

1

2

3

Objective No.

(b) MATLAB

2 0
1 0
0

0
4

3 0

(c) Latex

4

1

2

3

4

O b je c tiv e N o .

(d) Origin

Figure 23: An example of parallel coordinates plots in MS Excel, MATLAB, Latex, and Origin.
TABLE II
C REATING A PARALLEL COORDINATES PLOT IN MATLAB.

X = [15 31 20 50; 10 18 2 30; 20 5 32 20];
groups = {’a’, ’b’, ’c’};
parallelcoords(X,’group’,groups);
xlabel(’Objective No.’);

14
TABLE III
C REATING A PARALLEL COORDINATES PLOT IN L ATEX WITH PGFP LOTS
PACKAGE .

\usepackagepgfplots
\begintikzpicture
\beginaxis[xlabel=Objective No.,ylabel=Objective Values,style=font=,symbolic x coords=1, 2, 3, 4,xtick=data]
\addplot+[mark=none,draw=green,sharp plot] plot coor-

so that people could see some meaningful patterns (such as the examples in Figures 21 and 22). Another thought is to consider the coverage of lines between objectives in a parallel coordinates plot;
people may acquire more information from less coverage of the lines, for example, after exchanging
the order of objectives f1 and f2 in Figure 8.

Acknowledgement
This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC)
of U.K. under Grants EP/K001523/1 and EP/J017515/1, and National Natural Science Foundation
of China (NSFC) under Grants 61329302 and 61403326. X. Yao was also supported by a Royal
Society Wolfson Research Merit Award.

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