IEEE Computational Intelligence Magazine - August 2019 - 65

1

0.5

0.5

f3

f3

1

0
0

0
0
f1

0.5

0.5
f2

1 0

1

f1

0.5

1

0.5

0.5

f3

f3

1

(b)

1

0
0
f1

0.5
f2

1 0

(a)

0
0
0.5

0.5
f2

1 0

1

f1

0.5

0.5
f2

1 0

(c)

1

(d)

f2

FIGURE 3 The four solution sets given in Fig. 2 after being projected to a unit simplex. (a) Solution set P1, (b) solution set P2, (c) solution set P3 and (d) solution set P4.

f2

where p i denotes the i-th objective
value of p and M is the number of
objectives. Afterwards, all the points in P l
and R are projected to a unit simplex by
the following three steps, i.e., projection,
translation, and normalization:
1) Fo r e a c h p o i n t p i n P l a n d
R, p i = p i + (1/M ) - (1/M ) R Mj = 1 p j,
i = 1, f, M;
2) For each point p in P l and R,
p i = p i - min r ! R ri, i = 1, f, M;
3) For each point p in P l and R,
p i = p i /R Mj = 1 p j, i = 1, f, M.
In this way, all the points in P l and
R are located on the unit simplex
f1 + g + fM = 1 and within [0, 1] M .
Taking the solution sets shown in Fig. 2
as an example, the solution sets after
being projected to the unit simplex are
presented in Fig. 3, where the reference
point set R is the same to that used in
the IGD calculation in Table I.
It is worth noting that some MOEAs
also assess the diversity by projecting the
solutions to a simplex [28], [29], but
these methods are different from that in
the proposed CPF. Specifically, the proposed CPF projects each point p to its
foot drawn to the simplex, whereas
those MOEAs project each point p to
the intersection of op and the simplex.
In this way, the projection method in
CPF fixes the parallel distances between
points, whereas the projection method
in existing MOEAs fixes the angles
between points. As shown in Fig. 4, the
points projected by CPF have a similar
distribution to the original points; by
contrast, the distribution of the points is
highly distorted after being projected by
existing MOEAs, where the points in
the middle of the simplex become more
crowded than those on the border. As
illustration, Table II lists the Spacing values of the four solution sets shown in
Fig. 2 as well as those projected by the
methods in CPF and existing MOEAs.
It can be seen that P2 has the best evenness (i.e., smallest Spacing value), which
is followed by P1, P4 and P3. After
being projected by CPF, the rank of the
four solution sets in terms of Spacing is
not changed. But if the solution sets are
projected by existing MOEAs, P4 and
P1 will have the best and the worst

o

o

f1

f1

(a)

(b)
Original Points

Projected Points

FIGURE 4 The projection methods in CPF and some existing MOEAs. (a) CPF and (b) existing
MOEAs.

TABLE II The Spacing values of the four solution sets shown in Fig. 2 and those
projected by the methods in CPF and existing MOEAs.
PROJECTION
METHOD

P1

P2

P3

P4

ASCENDING
ORDER

ORIGINAL

2.42e-2

4.46e-3

5.62e-2

5.25e-2

P2, P1, P4, P3

CPF

1.46e-2

2.56e-3

3.25e-2

2.65e-2

P2, P1, P4, P3

EXISTING
MOEAs

3.48e-2

8.37e-3

3.45e-2

7.71e-7

P4, P2, P3, P1

AUGUST 2019 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE

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IEEE Computational Intelligence Magazine - August 2019

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