IEEE Computational Intelligence Magazine - August 2019 - 67

Vol = 3.9703e-01

0

0

Vol = 4.9094e-02

f2

1

f2

1

1

f1

0

0

(a)

0

0

f1

1

(b)

Vol = 2.0352e-01

Vol = 3.8653e-01

1

f2

1

D. Similarity and Difference Between
CPF and Other Metrics

1) CPF vs. Spacing: Spacing calculates
the distances between solutions,
which can only assess the evenness
of a solution set. By contrast, the
proposed CPF calculates the "volume" of a solution set, which can
assess both evenness and spread.
2) CPF vs. CL n: Both CL n and CPF
are based on a number of hypercubes. However, the CL n counts the
number of hypercubes having at
least one solution, which mainly
assesses the spread of a solution set.
Moreover, according to the definition of CL n, a solution set with
fewer solutions may have a better
metric value. By contrast, due to the
third property of monopolized
hypercube, the diversity of two
solution sets with different sizes is
comparable when using CPF.
3) CPF vs. PD: Both PD and the
monopolized hypercube in CPF are
designed for assessing the diversity of

by the example shown in Fig. 7,
even for two solution sets with similar convergence, PD may misjudge
their diversity due to the outlier in
the solution set. In contrast to PD,
the monopolized hypercube is not
influenced by the outlier due to the
restriction of the size length.

a point set filling in a unit hypercube. As mentioned before, the
diversity assessment in PD may be
influenced by the convergence of a
solution set, whereas CPF can eliminate such an influence by projecting
the solution set to a lower-dimensional space. In addition, as illustrated

f2

ratios of the "volumes" of the four solution
sets to the "volume" of the reference point
set, are 7.2193e-1, 8.9269e-2, 3.7006e-1
and 7.0283e-1, respectively, where the
CPF value of P1 is better than the others
and thus correctly reflects the diversity of
the four solution sets. It is worth noting
that the CPF value of any solution set is
within [0, 1] and not monotonic with
the number of solutions, which means
that a solution set to which some solution have been added may have smaller
CPF value than the same solution set
without the additional solutions. Therefore, if the density of the solution set is
considered as an important property, an
additional metric should be adopted to
take it into account.
Algorithm 1 summarizes the procedure of the proposed CPF. As can be
seen, the time complexity of CPF is
mainly determined by the calculation of
the "volume" of R, since there are a
much larger number of points in R than
in P. According to (7), the time complexity of CPF is O (MN 2 ), where M
and N denote the number of objectives
and reference points in R, respectively.

1

f1
(c)

0

0

f1

1

(d)

FIGURE 6 The monopolized hypercubes of the solutions given in Fig. 2, where the "volume"
(i.e., diversity) of each solution set is defined as the summation of the volumes of all the
monopolized hypercubes. (a) Solution set P1, (b) solution set P2, (c) solution set P3 and
(d) solution set P4.

Algorithm 1 Procedure of the proposed CPF.

1
2
3
4
5
6
7
8
9
10
11
12

Input: P (solution set), R (reference point set)
Output: CPF (CPF value of P )
Pl ! Replace each solution in P by its closest reference point in R;
Normalize the points in Pl and R by (5);
Project the points in Pl and R to a unit simplex;
Project the points in Pl and R to a unit hypercube;
Calculate the side length of each monopolized hypercube in R by (7);
Shrink the size length of each monopolized hypercube in R by (10);
Vol (R) ! Calculate the "volume" of R by (8);
Calculate the side length of each monopolized hypercube in Pl by (7);
Restrict the side length of each monopolized hypercube in Pl by (9);
Shrink the size length of each monopolized hypercube in Pl by (10);
Vol (P l ) ! Calculate the "volume" of Pl by (8);
CPF ! Vol (P l )/Vol (R);

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IEEE Computational Intelligence Magazine - August 2019

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