Evolutionary Multitasking for Multiobjective
Continuous Optimization: Benchmark Problems,
Performance Metrics and Baseline Results
Yuan Yuan∗ , Yew-Soon Ong∗ , Liang Feng† , A.K. Qin‡ , Abhishek Gupta∗ , Bingshui Da∗ ,
Qingfu Zhang§ , Kay Chen Tan¶ , Yaochu Jink , and Hisao Ishibuchi∗∗

arXiv:1706.02766v1 [] 8 Jun 2017

∗

School of Computer Science and Engineering, Nanyang Technological University, Singapore
† College of Computer Science, Chongqing University, China
‡ School of Science, RMIT University, Australia
§ Department of Computer Science, City University of Hong Kong, Hong Kong
¶ Department of Electrical and Computer Engineering, National University of Singapore, Singapore
k Department of Computer Science, University of Surrey, United Kingdom
∗∗ Department of Computer Science and Intelligent Systems, Osaka Prefecture University, Japan
Technical Report

I. I NTRODUCTION
With the explosion in the variety and volume of incoming
information streams, it is very desirable that the intelligent
systems or algorithms are capable of efficient multitasking.
The evolutionary algorithms (EAs) are a kind of populationbased search algorithms, which have the inherent ability to
handle multiple optimization tasks at once. By exploiting
this characteristic of EAs, evolutionary multitasking [1], [2]
has become a new paradigm in evolutionary computation
(EC), which signifies a multitasking search involving multiple
optimization tasks at a time, with each task contributing a
unique factor influencing the evolution of a single population
of individuals. The first tutorial on evolutionary multitasking
(“Evolutionary Multitasking and Implications for Cloud Computing”) was presented in IEEE Congress on Evolutionary
Computation (CEC) 2015, Sendai, Japan. Since then, a series
of studies on this topic have been conducted from various perspectives, which not only show the potentials of evolutionary
multitasking [3]–[8] but also explain why this paradigm works
[9], [10] to some extent.
Evolutionary multiobjective optimization (EMO) [11]–[13]
has always been a popular topic in the field of EC over the past
20 years. The reason mainly lies in the following two aspects.
On the one hand, many optimization problems in real-world
applications can be formulated as multiobjective optimization
problems (MOPs) in essence. On the other hand, EAs are able
to approximate the whole Pareto set (PS) or Pareto front (PF)
of a MOP in a single run due to the population-based nature,
which are regarded as quite suitable for solving MOPs. Until
now, plenty of EMO techniques have been developed, and
they can be roughly classified into three categories: Pareto
dominance-based algorithms [14]–[16], decomposition-based
algorithms [17]–[20], and indicator-based algorithms [21]–
[23].

Since evolutionary multitasking has shown promise in solving multiple tasks simultaneously and EMO is one of the
focused topics in EC, there may be noteworthy implications
of understanding the behavior of algorithms at the intersection
of the two paradigms. The combination of the two different
paradigms is referred to herein as multiobjective multifactorial
optimization (MO-MFO), which means that a number of
multiobjective optimization tasks are solved simultaneously by
evolving a single population of individuals using EAs. MOMFO has been first discussed in [4], where proof of concept
results from two synthetic benchmarks and a real-world case
study were presented.
In this report, we suggest nine test problems for MO-MFO,
each of which consists of two multiobjective optimization
tasks that need to be solved simultaneously. The relationship
between tasks varies between different test problems, which
would be helpful to have a comprehensive evaluation of the
MO-MFO algorithms. It is expected that the proposed test
problems will germinate progress the field of the MO-MFO
research.
The rest of this report is outlined as follows. Section II
presents the definitions of the proposed test problems. Section
III describes how to evaluate the performance of the algorithm
on the test problems. Section IV provides the baseline results
obtained by a MO-MFO algorithm (i.e., MO-MFEA [4])
and its underlying basic MOEA (i.e., NSGA-II [16]) on the
proposed benchmark problems.
II. D EFINITIONS OF B ENCHMARK P ROBLEMS
In the previous studies on evolutionary (single-objective)
multitasking [1], [2], [9], it is found that the degree of the
intersection of the global optima and the similarity in the
fitness landscape are two important ingredients that lead to the
complementarity between different optimization tasks. Motivated by this, the characteristics of multiobjective optimization

tasks in the test problems to be presented are focused on the
q(x) function, and the relationship between two multiobjective
optimization tasks in a test problem can be translated into the
relationship between their respective q(x) functions, which
is based on the fact that the Pareto optimal solutions of a
multiobjective task are achieved iff. its corresponding q(x)
reaches the global minimum.
Suppose that x∗1 and x∗2 are the global minima of q(x) functions in the two multiobjective tasks T1 and T2 , respectively.
Each dimension of x∗1 and x∗2 is then normalized to the same
range [0, 1], obtaining x̄∗1 and x̄∗2 . If x̄∗1 = x̄∗2 , we say that
the global minima are complete intersection; if there exists no
dimension where the values of x̄∗1 and x̄∗2 are equal, we call
that no intersection; all the other relationships between x̄∗1 and
x̄∗2 are referred to as partial intersection.
For computing the similarity between the fitness landscape
of q(x) functions, 1,000,000 points are randomly sampled in
the unified search space [1], then the Spearman’s rank correlation coefficient between q(x) is calculated as the similarity.
The similarity lying in (0, 1/3], (1/3, 2/3], (2/3, 1] is regarded
as high, medium, and low, respectively.
We consider three degrees of the intersection of the global
minima, i.e., complete, partial and no intersection, and within
each category, three categories of similarity in the fitness
landscape, i.e., high, medium, and low similarity. Accordingly,
there are nine test problems in total. Note that many practical settings give rise to a third condition for categorizing
potential multitask optimization settings, namely, based on the
phenotypic overlap of the decision variables [2]. To elaborate,
a pair of variables from distinct tasks may bear the same
semantic (or contextual) meaning, which leads to the scope
of knowledge transfer between them. However, due to the
lack of substantial contextual meaning in the case of synthetic
benchmark functions, such a condition for describing the
similarity/overlap between tasks is not applied in this technical
report.
The definitions of the proposed test problems are described in detail as follows. Note that there are five shift
vectors (scm2 , sph2 , spm1 , spl2 , snl1 ) and four rotation matrixes
(Mcm2 , Mpm1 , Mpm2 , Mnm2 ) involved, whose data can be
available online1 :
1) Complete Intersection with High Similarity (CIHS)
The first multiobjective task T1 is defined as follows:
πx1
),
2
πx1
min f2 (x) = q(x) sin(
),
2
n
X
q(x) = 1 +
x2i ,
min f1 (x) = q(x) cos(

The second multiobjective task T2 is defined as follows:
min f1 (x) = x1 ,
min f2 (x) = q(x)(1 − (

x1 2
) ),
q(x)

n

q(x) = 1 +

9 X
|xi |
n − 1 i=2

x1 ∈ [0, 1], xi ∈ [−100, 100], i = 2, 3, . . . , n
For this test problem, the number of decision variables n is
set to 50 for both T1 and T2 , and the similarity between T1
and T2 is 0.97. The PS and PF of T1 are given as follows:
P S : x1 ∈ [0, 1], xi = 0, i = 2, 3, . . . , 50
P F : f12 + f22 = 1, f1 ≥ 0, f2 ≥ 0

i=2

x1 ∈ [0, 1], xi ∈ [−100, 100], i = 2, 3, . . . , n
1 The data of rotation matrixes and shift vectors in the test problems can
be downloaded from
https://drive.google.com/open?id=0B8WAZ9HjQsUSdUY5UzBLN0NPd2M.

(3)

The PS and PF of T2 are given as follows:
P S : x1 ∈ [0, 1], xi = 0, i = 2, 3, . . . , 50
P F : f2 = 1 − f12 , 0 ≤ f1 ≤ 1

(4)

2) Complete Intersection with Medium Similarity (CIMS)
The first multiobjective task T1 is defined as follows:
min f1 (x) = x1 ,
min f2 (x) = q(x)(1 − (
q(x) = 1 +

x1 2
) ),
q(x)

n−1
X

(5)

(100(x2i − xi+1 )2 + (1 − xi )2 ),

i=2

x1 ∈ [0, 1], xi ∈ [−5, 5], i = 2, 3, . . . , n
The second multiobjective task T2 is defined as follows:
πx1
),
2
πx1
min f2 (x) = q(x) sin(
),
2
n
9 X
q(x) = 1 +
|zi |,
n − 1 i=2
min f1 (x) = q(x) cos(

(6)

(z2 , z3 , . . . zn )T = Mcm2 ((x2 , x3 , . . . , xn )T − scm2 ),
x1 ∈ [0, 1], xi ∈ [−5, 5], i = 2, 3, . . . , n
For this test problem, the number of decision variables n is
set to 10 for both T1 and T2 , and the similarity between T1
and T2 is 0.52. The PS and PF of T1 are given as follows:
P S : x1 ∈ [0, 1], xi = 1, i = 2, 3, . . . , 10

(1)

(2)

P F : f2 = 1 − f12 , 0 ≤ f1 ≤ 1

(7)

The PS and PF of T2 are given as follows:
P S : x1 ∈ [0, 1], (x2 , x3 , . . . , x10 )T = scm2
P F : f12 + f22 = 1, f1 ≥ 0, f2 ≥ 0

(8)

3) Complete Intersection with Low Similarity (CILS)

The first multiobjective task T1 is defined as follows:
πx1
),
2
πx1
min f2 (x) = q(x) sin(
),
2
n
X
q(x) = 1 +
(x2i − 10 cos(2πxi ) + 10)

For this test problem, the number of decision variables n is
set to 50 for both T1 and T2 , and the similarity between T1
and T2 is 0.99. The PS and PF of T1 are given as follows:

min f1 (x) = q(x) cos(

(9)

P S : x1 ∈ [0, 1], xi = 0, i = 2, 3, . . . , 50
p
P F : f2 = 1 − f1 , 0 ≤ f1 ≤ 1

(15)

The PS and PF of T2 are given as follows:

i=2

x1 ∈ [0, 1], xi ∈ [−2, 2], i = 2, 3, . . . , n
The second multiobjective task T2 is defined as follows:

P S : x1 ∈ [0, 1], (x2 , x3 , . . . , x50 )T = sph2
p
P F : f2 = 1 − f1 , 0 ≤ f1 ≤ 1

(16)

min f1 (x) = x1 ,
r

x1
),
q(x)
v
u
n
u 1 X
t
x2 )
q(x) = 21 + e − 20exp(−0.2
n − 1 i=2 i
min f2 (x) = q(x)(1 −

(10)

n

− exp(

1 X
cos(2πxi ))
n − 1 i=2

i=2
(z2 , z3 , . . . zn )T

x1 ∈ [0, 1], xi ∈ [−1, 1], i = 2, 3, . . . , n
For this test problem, the number of decision variables n is
set to 50 for both T1 and T2 , and the similarity between T1
and T2 is 0.07. The PS and PF of T1 are given as follows:
P S : x1 ∈ [0, 1], xi = 0, i = 2, 3, . . . , 50
P F : f12 + f22 = 1, f1 ≥ 0, f2 ≥ 0

(11)

q(x) = 1 +

The second multiobjective task T2 is defined as follows:

min f2 (x) = q(x)(1 − (
n
X

x1 2
) ),
q(x)

(zi2 − 10 cos(2πzi ) + 10),

(z2 , z3 , . . . zn )T = Mpm2 (x2 , x3 , . . . , xn )T ,
xi ∈ [0, 1], i = 1, 2, 3, . . . , n
For this test problem, the number of decision variables n is
set to 50 for both T1 and T2 , and the similarity between T1
and T2 is 0.55. The PS and PF of T1 are given as follows:
P S : x1 ∈ [0, 1], (x2 , x3 , . . . , x50 )T = spm1

r

(18)

i=2

(12)

min f1 (x) = x1 ,

n
X

xi ∈ [0, 1], i = 1, 2, 3, . . . , n

q(x) = 1 +

4) Partial Intersection with High Similarity (PIHS)
The first multiobjective task T1 is defined as follows:

min f2 (x) = q(x)(1 −

= Mpm1 ((x2 , x3 , . . . , xn )T − spm1 ),

min f1 (x) = x1 ,

The PS and PF of T2 are given as follows:
P S : x1 ∈ [0, 1], xi = 0, i = 2, 3, . . . , 50
p
P F : f2 = 1 − f1 , 0 ≤ f1 ≤ 1

5) Partial Intersection with Medium Similarity (PIMS)
The first multiobjective task T1 is defined as follows:
πx1
min f1 (x) = q(x) cos(
),
2
πx1
min f2 (x) = q(x) sin(
),
2
n
X
(17)
q(x) = 1 +
zi2

P F : f12 + f22 = 1, f1 ≥ 0, f2 ≥ 0

x1
),
q(x)
(13)

The PS and PF of T2 are given as follows:
P S : x1 ∈ [0, 1], xi = 0, i = 2, 3, . . . , 50

x2i ,

P F : f2 = 1 − f12 , 0 ≤ f1 ≤ 1

i=2

(19)

(20)

x1 ∈ [0, 1], xi ∈ [−100, 100], i = 2, 3, . . . , n
The second multiobjective task T2 is defined as follows:
min f1 (x) = x1 ,
min f2 (x) = q(x)(1 −
q(x) = 1 +

r

x1
),
q(x)

n
X
(zi2 − 10 cos(2πzi ) + 10),
i=2

(z2 , z3 , . . . zn )T = (x2 , x3 , . . . , xn )T − sph2 ,
x1 ∈ [0, 1], xi ∈ [−100, 100], i = 2, 3, . . . , n

(14)

6) Partial Intersection with Low Similarity (PILS)
The first multiobjective task T1 is defined as follows:
πx1
min f1 (x) = q(x) cos(
),
2
πx1
min f2 (x) = q(x) sin(
),
2
(21)
n
n
1 X 2 Y
xi
q(x) = 2 +
xi −
cos ( √
)
4000 i=2
i−1
i=2
x1 ∈ [0, 1], xi ∈ [−50, 50], i = 2, 3, . . . , n

The second multiobjective task T2 is defined as follows:
πx1
min f1 (x) = q(x) cos(
),
2
πx1
min f2 (x) = q(x) sin(
),
2
v
u
n
u 1 X
z2)
q(x) = 21 + e − 20exp(−0.2t
n − 1 i=2 i
(22)
n

1 X
− exp(
cos(2πzi ))
n − 1 i=2

For this test problem, the number of decision variables n is
set to 50 for both T1 and T2 , and the similarity between T1
and T2 is 0.002. The PS and PF of T1 are given as follows:
(23)

The PS and PF of T2 are given as follows:
P S : x1 ∈ [0, 1], (x2 , x3 , . . . , x50 )T = spl2
P F : f12 + f22 = 1, f1 ≥ 0, f2 ≥ 0

(29)

x1 ∈ [0, 1], x2 ∈ [0, 1], xi ∈ [−20, 20], i = 3, 4, . . . , n

x1 ∈ [0, 1], xi ∈ [−100, 100], i = 2, 3, . . . , n

P F : f12 + f22 = 1, f1 ≥ 0, f2 ≥ 0

πx1
πx2
) cos(
),
2
2
πx1
πx2
min f2 (x) = q(x) cos(
) sin(
),
2
2
πx1
min f3 (x) = q(x) sin(
),
2
n−1
X
q(x) = 1 +
(100(x2i − xi+1 )2 + (1 − xi )2 ),
min f1 (x) = q(x) cos(

i=3

(z2 , z3 , . . . zn )T = (x2 , x3 , . . . , xn )T − spl2 ,

P S : x1 ∈ [0, 1], xi = 0, i = 2, 3, . . . , 50

8) No Intersection with Medium Similarity (NIMS)
The first multiobjective task T1 is defined as follows:

(24)

The second multiobjective task T2 is defined as follows:
1
(x1 + x2 ),
2
x1 + x2 2
) ),
min f2 (x) = q(x)(1 − (
2 · q(x)
n
X
q(x) = 1 +
zi2
min f1 (x) =

(30)

i=3

(z3 , z4 , . . . zn )T = Mnm2 (x3 , x4 , . . . , xn )T ,
x1 ∈ [0, 1], x2 ∈ [0, 1], xi ∈ [−20, 20], i = 3, 4, . . . , n

7) No Intersection with High Similarity (NIHS)
The first multiobjective task T1 is defined as follows:
πx1
),
min f1 (x) = q(x) cos(
2
πx1
min f2 (x) = q(x) sin(
),
2
(25)
n−1
X
2
2
2
(100(xi − xi+1 ) + (1 − xi ) ),
q(x) = 1 +
i=2

For this test problem, the number of decision variables n is
set to 20 for both T1 and T2 , and the similarity between T1
and T2 is 0.51. The PS and PF of T1 are given as follows:
P S : x1 ∈ [0, 1], x2 ∈ [0, 1], xi = 1, i = 3, 4, . . . , 20
P F : f12 + f22 + f32 = 1, f1 ≥ 0, f2 ≥ 0, f3 ≥ 0

(31)

xi ∈ [0, 1], xi ∈ [−80, 80], i = 2, 3, . . . , n
The second multiobjective task T2 is defined as follows:
min f1 (x) = x1 ,
min f2 (x) = q(x)(1 −
q(x) = 1 +

n
X

r

The PS and PF of T2 are given as follows:
P S : x1 ∈ [0, 1], x2 ∈ [0, 1], xi = 0, i = 3, 4, . . . , 20

x1
),
q(x)

P F : f2 = 1 − f12 , 0 ≤ f1 ≤ 1

(32)

(26)
x2i ,

i=2

x1 ∈ [0, 1], xi ∈ [−80, 80], i = 2, 3, . . . , n
For this test problem, the number of decision variables n is
set to 50 for both T1 and T2 , and the similarity between T1
and T2 is 0.94. The PS and PF of T1 are given as follows:
P S : x1 ∈ [0, 1], xi = 1, i = 2, 3, . . . , 50
P F : f12 + f22 = 1, f1 ≥ 0, f2 ≥ 0

(27)

The PS and PF of T2 are given as follows:
P S : x1 ∈ [0, 1], xi = 0, i = 2, 3, . . . , 50
p
P F : f2 = 1 − f1 , 0 ≤ f1 ≤ 1

(28)

9) No Intersection with Low Similarity (NILS)
The first multiobjective task T1 is defined as follows:
πx2
πx1
) cos(
),
2
2
πx1
πx2
min f2 (x) = q(x) cos(
) sin(
),
2
2
πx1
min f3 (x) = q(x) sin(
),
2
n
n
1 X 2 Y
zi
q(x) = 2 +
zi −
cos ( √
)
4000 i=3
i
−2
i=3
min f1 (x) = q(x) cos(

(z3 , z4 , . . . zn )T = (x3 , x4 , . . . , xn )T − snl1 ,
x1 ∈ [0, 1], x2 ∈ [0, 1], xi ∈ [−50, 50], i = 3, 4, . . . , n

(33)

The second multiobjective task T2 is defined as follows:
1
(x1 + x2 ),
2
x1 + x2 2
min f2 (x) = q(x)(1 − (
) ),
2 · q(x)
v
u
n
u 1 X
q(x) = 21 + e − 20exp(−0.2t
x2 )
n − 2 i=3 i
min f1 (x) =

A. Performance Metric

n

− exp(

1 X
cos(2πxi ))
n − 2 i=3

x1 ∈ [0, 1], x2 ∈ [0, 1], xi ∈ [−100, 100], i = 3, 4, . . . , n
(34)
For this test problem, the number of decision variables n is
set to 25 for T1 and 50 for T2 , and the similarity between T1
and T2 is 0.001. The PS and PF of T1 are given as follows:
T

P S : x1 ∈ [0, 1], x2 ∈ [0, 1], (x3 , x4 , . . . , x25 ) = snl1

(35)

P F : f12 + f22 + f32 = 1, f1 ≥ 0, f2 ≥ 0, f3 ≥ 0
The PS and PF of T2 are given as follows:
P S : x1 ∈ [0, 1], x2 ∈ [0, 1], xi = 0, i = 3, 4, . . . , 50

(36)

P F : f2 = 1 − f12 , 0 ≤ f1 ≤ 1

Table I summarizes the proposed test problems together
with their properties, where sim(T1 , T2 ) denotes the similarity
between two tasks in a test problem. Fig. 1 shows four kinds
of the Pareto fronts of the multiobjective optimization tasks in
the test problems.
   

1


      

          

1

0.6

0.6



0.8



0.8

0.4

0.4

0.2

0.2

0

0
0

1

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

problems. All the test problems should be treated as black-box
problems, i.e., the analytic forms of these problems are not
known for the algorithms.

0.8

1





          

               

The inverted generational distance (IGD) [24] is used to
evaluate the performance of an algorithm on each task of the
considered test problem. Let A be a set of nondominated
objective vectors that are obtained for a task Ti by the
algorithm, and P ∗ be a set of uniformly distributed objective
vectors over the PF of Ti . A and P ∗ are first normalized using
the maximum and minimum objective values among P ∗ , then
the metric IGD of the approximate set A is calculated as:
sX
1
(min d(x, y))2
(37)
IGD(A, P ∗ ) = ∗
y∈A
|P |
∗
x∈P

where d(x, y) is the Euclidean distance between x and y in the
normalized objective space. If |P ∗ | is large enough to represent
the PF, the IGD(A, P ∗ ) could measure both convergence and
diversity of A in a sense. A small value of IGD(A, P ∗ ) means
that A must be very close to P ∗ and cannot miss any part of
P ∗.
The data file of P ∗ for each task and the source code of
computing IGD can also be available from the zip package
provided online.
B. Ranking of the MO-MFO Algorithms
For a multiobjective multitasking problem, several IGD
values are obtained by each run of the algorithm, and one
value is for a task. Based on these IGD values, we provide
a comprehensive criterion to evaluate the overall performance
of a MO-MFO algorithm, so as to rank the algorithms on each
test problem.
Suppose there are K tasks T1 , T2 , . . . , TK in a test problem,
and the IGD value obtained by an algorithm in a run for Ti
is denoted as Ii , where i = 1, 2, . . . K. Moreover, suppose
the average and standard deviation of the IGD values for Ti
are µi and σi respectively, where i = 1, 2, . . . , K. The mean
standard score (MSS) of the obtained IGD values for the test
problem is computed as follows:
M SS =

1

0.8





0.6
0.5

0.4
0
1

0.2

1
0.5

0.5

0
0

0.2

0.4

0.6

0.8

1



0

0





K
1 X Ii − µi
K i=1 σi

(38)

In practice, µi and σi will be calculated according to the IGD
results obtained by all the algorithms on the task Ti in all
runs. MSS is used as a comprehensive criterion, and a smaller
MSS value indicates better overall performance of a MO-MFO
algorithm on a test problem.
C. Experimental Settings

Fig. 1. Four different kinds of Pareto fronts involved in the proposed test
problems.

III. P ERFORMANCE E VALUATION
This section describes the process that will be followed to
evaluate the performance of the algorithm on the proposed test

1) The maximum number of nondominated objective vectors
in the approximate set: After running the algorithm on a
test problem, the IGD metric should be computed for each
task of the problem. The maximum number of nondominated
objective vectors produced by the algorithm for computing the
IGD (i.e., |A|) should be:

TABLE I
S UMMARY OF THE PROPOSED TEST PROBLEMS FOR E VOLUTIONARY M ULTIOBJECTIVE M ULTITASKING .
Problem

CIHS

CIMS

CILS

PIHS

PIMS

PILS

NIHS

NIMS

NILS

•
•

sim
(T1 , T2 )

Task
No.

Pareto Set

Pareto Front

Properties

T1

x1 ∈ [0, 1],
xi = 0, i = 2 : 50

f12 + f22 = 1,
f1 ≥ 0, f2 ≥ 0

concave, unimodal,
separable

T2

x1 ∈ [0, 1],
xi = 0, i = 2 : 50

f2 = 1 − f12 ,
0 ≤ f1 ≤ 1

concave, unimodal,
separable

T1

x1 ∈ [0, 1],
xi = 1, i = 2 : 10

f2 = 1 − f12 ,
0 ≤ f1 ≤ 1

concave, multimodal,
nonseparable

T2

x1 ∈ [0, 1],
(x2 , . . . , x10 )T = scm2

f12 + f22 = 1,
f1 ≥ 0, f2 ≥ 0

concave, unimodal,
nonseparable

T1

x1 ∈ [0, 1],
xi = 0, i = 2 : 50

concave, multimodal,
separable

T2

x1 ∈ [0, 1],
xi = 0, i = 2 : 50

T1

x1 ∈ [0, 1],
xi = 0, i = 2 : 50

T2

x1 ∈ [0, 1],
(x2 , . . . , x50 )T = sph2

f12 + f22 = 1,
f1 ≥ 0, f2 ≥ 0
√
f 2 = 1 − f1 ,
0 ≤ f1 ≤ 1
√
f 2 = 1 − f1 ,
0 ≤ f1 ≤ 1
√
f 2 = 1 − f1 ,
0 ≤ f1 ≤ 1

T1

x1 ∈ [0, 1],
(x2 , . . . , x50 )T = spm1

f12 + f22 = 1,
f1 ≥ 0, f2 ≥ 0

concave, unimodal,
nonseparable

T2

x1 ∈ [0, 1],
xi = 0, i = 2 : 50

f2 = 1 − f12 ,
0 ≤ f1 ≤ 1

concave, multimodal,
nonseparable

T1

x1 ∈ [0, 1],
xi = 0, i = 2 : 50

f12 + f22 = 1,
f1 ≥ 0, f2 ≥ 0

concave, multimodal,
nonseparable

T2

x1 ∈ [0, 1],
(x2 , . . . , x50 )T = spl2

f12 + f22 = 1,
f1 ≥ 0, f2 ≥ 0

concave, multimodal,
nonseparable

T1

x1 ∈ [0, 1],
xi = 1, i = 2 : 50

concave, multimodal,
nonseparable

T2

x1 ∈ [0, 1],
xi = 0, i = 2 : 50

T1

x1 ∈ [0, 1], x2 ∈ [0, 1],
xi = 1, i = 3 : 20

f12 + f22 = 1,
f1 ≥ 0, f2 ≥ 0
√
f 2 = 1 − f1 ,
0 ≤ f1 ≤ 1
P3
2
i=1 fi = 1,
fi ≥ 0, i = 1, 2, 3

T2

x1 ∈ [0, 1], x2 ∈ [0, 1],
xi = 0, i = 3 : 20

concave, unimodal,
nonseparable

T1

x1 ∈ [0, 1], x2 ∈ [0, 1],
(x3 , . . . , x25 )T = snl1

f2 = 1 − f12 ,
0 ≤ f1 ≤ 1
P3
2
i=1 fi = 1,
fi ≥ 0, i = 1, 2, 3

concave, multimodal,
nonseparable

T2

x1 ∈ [0, 1], x2 ∈ [0, 1],
xi = 0, i = 3 : 50

f2 = 1 − f12 ,
0 ≤ f1 ≤ 1

concave, multimodal,
nonseparable

0.97

0.52

0.07

0.99

0.55

0.002

0.94

0.51

0.001

100 for the task with two objectives.
120 for the task with three objectives.

2) The maximum number of function evaluations: It is set
to 200,000 for all the test problems. Note that “a function
evaluation” here means a calculation of the objective functions
of a task Ti , and the function evaluations on different tasks
are not distinguished.
3) The number of independent runs: Each algorithm should
be run 30 times independently on each multiobjective multitasking test problem.

convex, multimodal,
nonseparable
convex, unimodal,
separable
convex, multimodal,
separable

convex, unimodal,
separable
concave, multimodal,
nonseparable

4) The parameter settings for the algorithm: The test
problem NIMS or NILS consists of a bi-objective task and
a three-objective task. Whereas each of the remaining test
problems, consists of two bi-objective tasks. The algorithm
should use the same parameters for NIMS and NILS, and use
the same ones for the other test problems.
IV. BASELINE R ESULTS
This section presents the baseline results obtained by a
MO-MFO algorithm, referred to as MO-MFEA [4], on the
proposed test problems. MO-MFEA will be degenerated into

NSGA-II when there is only one task in the input. To show
the benefits of MO-MFO, the results of MO-MFEA are also
compared with those of NSGA-II. It is worth note that NSGAII is not a MO-MFO algorithm and it solves the two tasks in
a test problem separately, whereas MO-MFEA solves them
simultaneously in a single population.
NSGA-II uses a population size of 100 for solving a single
task, while MO-MFEA uses a population size of 200 for
solving a test problem with two tasks. The maximal number
of function evaluations on a task is set to 100,000 for NSGAII. To ensure a fair comparison, MO-MFEA uses 200,000
as the maximal number of function evaluations for a test
problem since it solves two tasks together at a time. NSGAII and MO-MFEA use the same simulated binary crossover
operator and polynomial mutation to produce the candidate
individuals. The parameters for crossover and mutation are
shown in Table II, where D is the dimensionality of the unified
code representation [1].
TABLE II
PARAMETERS FOR CROSSOVER AND MUTATION .
Parameter

Value

Crossover probability (pc )
Mutation probability (pm )
Distribution index for crossover (ηc )
Distribution index for mutation (ηm )

0.9
1/D
20
20

TABLE III
T HE AVERAGE AND STANDARD DEVIATION ( SHOWN IN THE BRACKET ) OF
IGD VALUES OBTAINED BY MO-MFEA AND NSGA-II. T HE
SIGNIFICANTLY BETTER IGD VALUE FOR EACH TASK IS HIGHLIGHTED IN
BOLD .
Problem

sim

Task

(T1 , T2 )

No.
T1

CIHS

0.97
T2
T1

CIMS

0.52
T2
T1

CILS

0.07
T2
T1

PIHS

0.99
T2
T1

PIMS

0.55
T2
T1

A. Results of IGD and MSS
Table III shows the average IGD obtained by MO-MFEA
and NSGA-II on nine test problems, where the standard
deviation values are also reported. To test the difference for
statistical significance, the Wilcoxon signed-rank test at a 5%
significance level is conducted on the IGD values obtained by
MO-MFEA and NSGA-II for each multiobjective task, and the
significantly better IGD value for each task is shown in bold.
From Table III, except for PILS and NILS, MO-MFEA
performs significantly better than NSGA-II on both the two
tasks in a test problem. For some cases, e.g., T1 of CILS
and T1 of NIHS, MO-MFEA even outperforms NSGA-II by a
large margin. For PILS, although MO-MFEA is a little worse
than NSGA-II on T1 , it performs much better than NSGA-II
on T2 . As for NILS, MO-MFEA and NSGA-II achieve very
close performance on both T1 and T2 .
Table IV further reports the difference between the average
MSS obtained by MO-MFEA and NSGA-II on each test
problem. Here µi and σi are calculated according to the IGD
results of MO-MFEA and NSGA-II for Ti in all runs. As can
be seen from Table IV, all the differences are smaller than
0 except on NILS, which indicates that MO-MFEA achieves
better overall performance than NSGA-II on all the proposed
test problems except NILS.
B. Convergence Curves
Fig. 2 shows the evolution of average IGD with number of
generations for MO-MFEA and NSGA-II on each of the test

PILS

0.002
T2
T1

NIHS

0.94
T2
T1

NIMS

0.51
T2
T1

NILS

0.001
T2

IGD
NSGA-II

MO-MFEA

2.0234E-3

3.9912E-4

(5.1879E-4)

(9.7671E-5)

4.3621E-3

2.6491E-3

(8.5407E-4)

(5.6744E-4)

1.0045E-1

4.5705E-2

(7.5847E-2)

(6.5713E-2)

2.2897E-2

8.7723E-3

(1.9082E-2)

(1.2175E-2)

2.5503E-1

2.7105E-4

(1.0821E-1)

(2.6731E-5)

1.9905E-4

1.8986E-4

(6.513E-6)

(6.628E-6)

1.1145E-3

1.1029E-3

(3.8107E-4)

(1.1949E-3)

5.654E-2

3.0406E-2

(3.0369E-2)

(1.6188E-2)

4.4903E-3

2.6206E-3

(1.6131E-3)

(1.1806E-3)

1.5577E1

1.0892E1

(3.7002E0)

(3.962E0)

2.7647E-4

3.2401E-4

(1.2629E-4)

(8.8287E-5)

6.3458E-1

1.099E-2

(8.8258E-4)

(2.0652E-3)

3.1286E1

1.5523E0

(6.0683E1)

(2.4312E-2)

7.8558E-4

5.0224E-4

(1.6655E-4)

(1.4331E-4)

4.7021E-1

2.792E-1

(3.3048E-1)

(2.6437E-1)

9.1944E-2

2.8576E-2

(8.8784E-2)

(4.7721E-2)

8.3588E-4

8.3483E-4

(5.4979E-5)

(6.2227E-5)

6.4226E-1

6.4316E-1

(2.5715E-4)

(3.2324E-4)

problem. From Fig. 2, MO-MFEA converges to the two Pareto
fronts (for T1 and T2 respectively) faster than NSGA-II on
most of test problems by solving the two multiobjective tasks
in a test problem simultaneously. For certain cases, NSGAII shows a little faster convergence speed in the early stage
of search but is surpassed by MO-MFEA quickly, which is
particularly clear on CILS.
C. Distribution of Solutions in the Objective Space
The distribution of nondominated objective vectors obtained
by MO-MFEA and NSGA-II in a single run for each task
is shown in Figs. 3, 4, and 5, in order to visually see how

IGD (log scale)

10 0

10 -2

10 2

10 0

200

400

600

800

10 -4

1000

0

200

 
 
 
 

IGD (log scale)

IGD (log scale)

600

800

10 -2

10 -4

1000

10 0

10 2

200

400

600

800

10 0

10 -4

1000

0

200

IGD (log scale)

10 5

400

600

800

10 0

200

400

600

800

1000

10 -2

0

200

400

600

800

1000

NILS

10 0
 
 
 
 

10 4

10 2

10 -2

1000

Number of Generations

10 0

0

800

 
 
 
 

10 -1

10 -4

1000

NIMS

10 6
 
 
 
 

600

PILS

Number of Generations

NIHS

400

10 -3

Number of Generations

10 10

200

10 0

 
 
 
 

10 -2

0

0

Number of Generations

PIMS

10 4

10 -2

IGD (log scale)

10 0

Number of Generations

PIHS

10 2

400

IGD (log scale)

0

10 4

10 -5

 
 
 
 

10 -2

Number of Generations

10 -4

CILS

10 2
 
 
 
 

IGD (log scale)

IGD (log scale)

10 2

10 -4

CIMS

10 4
 
 
 
 

IGD (log scale)

CIHS

10 4

 
 
 
 

10 -1

10 -2

10 -3

0

200

Number of Generations

400

600

800

Number of Generations

1000

10 -4

0

200

400

600

800

1000

Number of Generations

Fig. 2. Convergence curves of the average IGD (over 30 runs) obtained with the number of generations for MO-MFEA and NSGA-II on the proposed test
problems.

TABLE IV
T HE DIFFERENCE BETWEEN THE AVERAGE MSS OBTAINED BY
MO-MFEA AND NSGA-II.
Problem sim(T1 , T2 ) MSS Difference
COHS
COMS
COLS
POHS
POMS
POLS
NOHS
NOMS
NOLS

0.97
0.52
0.07
0.99
0.55
0.002
0.94
0.51
0.001

-1.672
-0.764
-1.43
-0.48
-1.073
-0.787
-1.001
-0.71
0.832

well the obtained approximate set is in the objective space.
This particular run is associated with the result closest to the
average IGD value. It can be seen from these figures that MOMFEA can achieve a better PF approximation than NSGA-II
almost on all the concerned multiobjective optimization tasks.
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1.5

5
1.5

1
1

0.5



0.5
0

0



0
0

0.2

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0.6

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