A Self-Adaptive Response Strategy for Dynamic
Multiobjective Evolutionary Optimization
Based on Objective Space Decomposition

Ruochen Liu
ruochenliu@xidian.edu.cn
Key Lab of Intelligent Perception and Image Understanding of Ministry of Education,
Xidian University, Xi’an, 710071, 中国

Jianxia Li
1761389691@qq.com
Key Lab of Intelligent Perception and Image Understanding of Ministry of Education,
Xidian University, Xi’an, 710071, 中国

Yaochu Jin
计算机科学系, University of Surrey, Guildford, GU2 7XH,
英国

yaochu.jin@surrey.ac.uk

Licheng Jiao
lcjiao@mail.xidian.edu.cn
Key Lab of Intelligent Perception and Image Understanding of Ministry of Education,
Xidian University, Xi’an, 710071, 中国

https://doi.org/10.1162/evco_a_00289

抽象的

Dynamic multiobjective optimization deals with simultaneous optimization of multi-
ple conflicting objectives that change over time. Several response strategies for dynamic
optimization have been proposed, which do not work well for all types of environmen-
tal changes. 在本文中, we propose a new dynamic multiobjective evolutionary algo-
rithm based on objective space decomposition, in which the maxi-min fitness function
is adopted for selection and a self-adaptive response strategy integrating a number of
different response strategies is designed to handle unknown environmental changes.
The self-adaptive response strategy can adaptively select one of the strategies accord-
ing to their contributions to the tracking performance in the previous environments.
Experimental results indicate that the proposed algorithm is competitive and promis-
ing for solving different DMOPs in the presence of unknown environmental changes.
同时, the proposed algorithm is applied to solve the parameter tuning problem
of a proportional integral derivative (PID) controller of a dynamic system, obtaining
better control effect.

关键词

Dynamic multiobjective optimization, objective space decomposition, maxi-min fitness
function, self-adaptive response strategy, PID control.

1

介绍

Over the past years, a number of evolutionary algorithms have been proposed to solve
multiobjective optimization problems (MOPs) that involve two or more conflicting ob-
jectives (Coello Coello, 2002; Deb et al., 2002; Fonseca and Fleming, 1993; Zhang and Li,
2007; 张等人。, 2008; Zitzler et al., 2002). Many MOPs in the real world, 然而,

稿件收到: 23 十二月 2019; 修改: 30 十月 2020 和 14 一月 2021; accepted: 18 一月
2021.
© 2021 麻省理工学院

进化计算 29(4): 491–519

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右. 刘等人.

are subject to various types of uncertainty, resulting in time-varying objective and / 或者
constraint functions (Deb et al., 2007). These MOPs are referred to as dynamic multiob-
jective optimizations problems (DMOPs). 最近几年, increasing attention has been
paid to solving DMOPs in the real world, such as path planning (Michalewicz et al.,
2007), evolutionary robotics (Tinós and Yang, 2007), and financial optimization prob-
莱姆斯 (Tezuka et al., 2007).

The goal of a dynamic multiobjective optimization algorithm is to track the chang-
ing Pareto-optimal set (聚苯乙烯) (Pelosi and Selleri, 2014). In order to efficiently solve
DMOPs, two basic requirements need to be considered. 第一的, the algorithm should be
able to effectively respond to environmental changes. 第二, the algorithm is able to
quickly find the PS of the current environment before the next change occurs. 最近,
a large number of dynamic multiobjective evolutionary algorithms have been proposed
(Jiang and Yang, 2017乙; Zhou et al., 2014, 2007), 哪个, 然而, usually perform well
only for specific types of environmental changes. To address this issue, this article pro-
poses a self-adaptive response strategy (SRS) that is able to adaptively select different
response strategies for different environmental changes according to the contribution of
each response strategy. We adopt a multiobjective evolutionary algorithm based on ob-
jective space decomposition (MOEA-OSD) as the basic optimizer. 所以, 整个
algorithm combining MOEA-OSD with SRS is called MOEA-OSD/SRS.

Note that the main new contribution of MOEA-OSD/SRS lies in SRS, which in-
tegrates several popular response strategies, including random diversity introduction
(RDI) (Deb et al., 2007), mutational diversity introduction (MDI) (Deb et al., 2007),
linear prediction strategy (LPS) (Zhou et al., 2007), feed-forward prediction strategy
(FPS) (Hatzakis and Wallace, 2006), and population prediction strategy (PPS) (周
等人。, 2014). By adding different labels, individuals generated by different response
strategies can be traced, and their contributions in the previous environments can be
记录的. Once the environment changes, the probability at which each response strat-
egy is selected in the new environment is determined according to its contribution in the
previous environments, making the proposed algorithm adaptive to unknown environ-
mental changes.

In order to quickly find the PS of the current environment before the next environ-
mental change occurs, MOEA-OSD is adopted as the basic static multiobjective evo-
lutionary algorithm. MOEA-OSD decomposes the objective space of the MOP into a
number of subobjective spaces using a set of uniformly distributed reference vectors.
The reference vectors are generated uniformly in the objective space, evenly dividing
the objective space into a number of subobjective spaces. The subobjective spaces are
treated as an external archive, and the corresponding historical optimal solutions of
each reference vector are stored in the corresponding subobjective space. Owing to the
evenly divided subobjective spaces, the algorithm can naturally maintain a good diver-
城市. 此外, the algorithm adopts the maxi-min fitness function (Balling, 2003) 到
calculate the fitness value of each solution, which is believed to be able to simultane-
ously account for both convergence and diversity of the solutions.

To examine the performance of MOEA-OSD/SRS, three sets of comprehensive em-
pirical studies are performed, including a comparison of SRS with other six effective
response strategies, a comparison of MOEA-OSD/SRS with other seven state-of-the-
art dynamic multiobjective optimization algorithms, and a comparison of MOEA-OSD
with other six multiobjective evolutionary algorithms. 同时, MOEA-OSD/SRS is
employed to tune the parameters of a proportional integral derivative (PID) controller
for a dynamic system.

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A Self-Adaptive Response Strategy for DMOPs

The rest of this article is organized as follows. 部分 2 presents a brief review
of response strategies considered in the current work. The proposed MOEA-OSD/SRS
is presented in detail in Section 3. 部分 4 introduces the benchmark functions, pa-
rameter settings, and performance metrics, followed by a detailed description of the
experimental results and analysis. 部分 5 applies MOEA-OSD/SRS to optimize the
parameters of a PID controller of a dynamic system. 最后, 部分 6 draws a conclu-
sion of the work and suggests future research.

2 Background

This section briefly discusses the related work on solving DMOPs and provides a de-
tailed review of the widely used response strategies.

一般来说, a dynamic multiobjective minimization problem can be described as

如下 (Farina et al., 2004):

⎧
⎪⎨

⎪⎩

min y = F (X, t ) = (f1(X, t ), f2(X, t ), . . . , fm(X, t ))

s.t.

gi (X, t ) ≤ 0,
hj (X, t ) = 0,

i = 1, 2, . . . . . . , p
j = 1, 2, . . . . . . , q,

(1)

where x = [x1, x2, . . . , xn] is the decision vector, and F (X, t ) is the set of objectives to
be minimized with respect to time t. Functions g(X, t ) and h(X, t ) represent inequality
and equality constraints, 分别. 在这项工作中, 然而, we focus only on uncon-
strained DMOPs.

Comprehensive reviews of research on solving DMOPs can be found in Cruz et al.
(2011), Goh and Tan (2009乙), Helbig and Engelbrecht (2014), Jin and Branke (2005),
Nguyen et al. (2012), and Raquel and Yao (2013). Evolutionary algorithms (Avdagi´c
等人。, 2009; Jiang and Yang, 2017乙; Wu et al., 2015; Zeng et al., 2006), particle swarm
optimization algorithms (Greeff and Engelbrecht, 2008, 2010; Helbig and Engelbrecht,
2011, 2012, 2013乙; Rong et al., 2018; Salazar Lechuga, 2009), and immune algorithms
(Shang et al., 2005, 2014) have been used to solve DMOPs. 此外, 合作社
co-evolutionary algorithms have been designed for dynamic multiobjective optimiza-
的 (Goh and Tan, 2009A; 刘, 陈等人。, 2014).

While they have shown better performance in solving static MOPs (Chang et al.,
2008; Ishibuchi et al., 2009; Li and Zhang, 2009; Liu and Niu, 2013; Yuen and Ramli,
2010; Zhang and Li, 2007), decomposition-based MOEAs have hardly been used to solve
DMOPs. To fill the gap, this work adopts a dynamic multiobjective optimization algo-
rithm based on objective space decomposition.

For a multiobjective optimization algorithm to be capable of solving DMOPs, 四
basic approaches to responding to environmental changes have been investigated.
These approaches are typically used for generating the initial population in the new en-
环境. We review the widely used DMOAs with respect to the adopted response
strategy in greater detail.

2.1 Diversity Introduction Strategy

When solving a DMOP, it becomes difficult for an optimization algorithm to quickly lo-
cate the global optimum usually due to the lack of diversity in the population when an
environmental change occurs. 因此, it is helpful to introduce diversity when environ-
mental changes are detected (科布, 1990; Deb et al., 2007; Goh and Tan, 2009A; Greeff
and Engelbrecht, 2008; Jin and Branke, 2005; 刘等人。, 2017; Vavak et al., 1997).

Evolutionary Computation Volume 29, 数字 4

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右. 刘等人.

Two typical diversity introduction strategies are proposed by Deb et al. (2007). 在
one of the approaches, a certain proportion of the individuals in the population are
replaced by randomly generated individuals, which is referred to as DNSGA-II-A. 在里面
other approach, a certain proportion of the individuals in the population are mutated,
known as DNSGA-II-B.

The diversity introduction strategy is simple and easy to implement, but it responds
to environmental changes fairly arbitrarily, which may mislead the evolutionary search.
同时, the proportion of diversity introduction has a great impact on the perfor-
mance of the algorithm.

2.2 Diversity Maintenance Strategy

Diversity maintenance strategies (Bui et al., 2005; Grefenstette et al., 1992; Morrison,
2002; Zeng et al., 2006) aim to maintain a certain degree of diversity of the population
in the process of evolutionary optimization. 例如, Zeng et al. (2006) proposed
a dynamic multiobjective evolutionary algorithm based on an orthogonal evolutionary
algorithm (DOMOEA) to solve continuous DMOPs. The idea is simply to adopt the
optimal solutions of the previous environment to initialize the population in the new
环境. This strategy is therefore computationally very efficient.

很遗憾, diversity maintenance strategies are mainly suited for solving con-
tinuous optimization problems. They are good for solving DMOPs with minor environ-
mental changes, but may perform poorly when the changes are severe.

2.3

Prediction-Based Strategy

If there is a certain relationship between the optimal solutions in the new environment
and those in the previous environments, it is desirable to predict the location of the new
optimal solutions and include them in the initial population for the new environment.
Hatzakis and Wallace (2006) proposed a forward-looking approach as the response
机制. Once an environmental change is detected, the position of the optimal solu-
tion in the new environment is predicted by using an autoregressive model. Zhou et al.
(2007) proposed a prediction-based reinitialization strategy (PRI). In PRI, two strate-
gies are introduced when an environmental change is detected. The first strategy is to
predict the new position of each individual in the population according to its previous
position change. The second strategy is to perturb the current population with Gaussian
noise. PRI was further extended in Zhou et al. (2014) by suggesting a prediction strat-
egy based on the whole population (PPS). In PPS, a Pareto set is divided into two parts:
a center point and a manifold. A set of center points in the previous environments are
used to estimate the center in the new environment, and previous manifolds are used
to predict the new manifold. 然后, the initial population in the new environment can
be generated by combining the predicted center and manifold. 最近, Jiang and Yang
(2017乙) developed a steady-state and generational evolutionary algorithm (SGEA) 那
can respond to changes in a steady-state manner. When an environmental change is de-
tected, SGEA reuses multiple outdated solutions with a better distribution and relocates
partial solutions to the regions near the new Pareto optimal front (PF) based on the in-
formation gathered from the previous and new environments. Other prediction-based
strategies have been reported (刘, 2010; Wu et al., 2015).

Prediction-based strategies work effectively if the prediction is sufficiently reliable.
很遗憾, inaccurate predictions may mislead the evolutionary search. 通常,
prediction-based strategies work more efficiently for DMOPs having periodic environ-
mental changes.

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A Self-Adaptive Response Strategy for DMOPs

2.4 Memory-Based Strategy

Last but not the least, memory-based strategies (Ng and Wong, 1995; 哪个, 2006; 张
and Qian, 2011) have also been designed to respond to environmental changes. 这些
strategies store relevant information about each of the previous environments and reuse
the stored information in the new environment. 例如, 哪个 (2006) proposed an
associative memory mechanism in which the solutions and a distribution estimation
are stored as memory individuals. When an environmental change is detected, all in-
dividuals in the memory are re-evaluated. The better an individual is, the more likely
its related distribution will be selected to generate the initial population in the new
环境.

Similar to the prediction-based strategies, memory-based strategies are useful for
solving DMOPs having periodic changes. 然而, storing information in and retriev-
ing information from the memory can be computationally intensive. Note that this strat-
egy is useful for solving periodic DMOPs only.

Almost all response strategies discussed above have both advantages and disad-
vantages and work well in dealing with certain types of environmental changes. 为了
this reason, we propose a self-adaptive response strategy for dynamic multiobjective
evolutionary algorithms, which is able to automatically select the most suited response
strategies for given problems.

3 The Proposed MOEA-OSD/SRS

As discussed before, a DMOA should be able to achieve a set of diverse Pareto optimal
solutions of the current environment before a new environmental change occurs. 在
添加, it should be able to timely detect an environmental change and respond to the
environmental change effectively.

To meet the above requirements, the proposed algorithm adopts MOEA-OSD, A
decomposition-based approach, as the base optimizer. A self-adaptive response strategy
is embedded in MOEA-OSD to quickly respond to unknown environmental changes. 在
this section, we describe MOEA-OSD, the environmental change detection mechanism,
and the self-adaptive response strategy (SRS), before the overall framework of MOEA-
OSD/SRS is presented.

3.1 MOEA-OSD

This section introduces the main principle of MOEA-OSD. In MOEA-OSD, firstly, a set
of uniformly distributed reference vectors (r 1, r 2, . . . , r N ) are generated to divide the
objective space into a number of subobjective spaces. Then MOEA-OSD will find the
closest solution of each reference vector, which will be introduced in the following sec-
的. In the evolutionary optimization process, the offspring population is produced
by performing the differential crossover operator (Price et al., 2006) and Gaussian mu-
tation operator (Higashi and Iba, 2003) on parent individuals, and then the parent and
offspring populations are merged to form a combined population. The maxi-min fitness
function is used to calculate the fitness value of all individuals, and the individuals with
smaller fitness value are selected to generate the next population. 最后, the closest so-
lution of each reference vector is updated. The above process repeats until a termination
criterion is satisfied. The main steps of MOEA-OSD are described in Algorithm 1.

Finding the Closest Solution of Each Reference Vector

3.1.1
In order to generate a set of uniformly distributed reference vectors, we adopt the ap-
proach introduced in Cheng et al. (2016).

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第一的, a set of uniformly distributed points are generated on a unit hyperplane, 作为

presented in Eq. (2):

ui = (u1

我 , u2

我 , . . . , um
我 )

(西德:6)

uj
我

∈

0
H

,

1
H

, . . . ,

H
H

(西德:7)

米(西德:8)

,

j =1

uj
我

= 1,

(2)

where i = 1, 2, . . . . . . , N with N being the number of uniformly distributed points, m is
the number of objectives, and H is a positive integer for the simplex-lattice design.

然后, the corresponding reference vectors can be obtained by Eq. (3):

(西德:4)ui(西德:4) .
For a set of uniformly distributed reference vectors (r 1, r 2, . . . , r N ) and a population

(3)

r i = ui

POP, each solution finds its closest reference vector in following way:

V i =

(西德:6)
r|(西德:2)(F (希 ), r ) = max
1≤j ≤N

(西德:7)

(西德:9)

(西德:10)

(西德:2)(F (希 ), r j )

xi ∈ POP, i = 1, 2, . . . . . . , 氮
(西德:2)(F (希 ), r j ) = r j × (F (希 ) − Z)时间
(西德:11)
(西德:11)
(西德:11) ×
(西德:11)r j
(F (希 ) − Z)
where V i represents the corresponding closest reference vector of the solution xi,
Z = (Z1, Z2, . . . , Zm) is the reference point, and Zk = min
for each
k = 1, 2, . . . . . . , m and (西德:2)(F (希 ), r j ) is the cosine of angle between the vector r j and
F (希 ) − Z.

fk (X)|x ∈ POP

(西德:11)
(西德:11) ,

(4)

(西德:11)
(西德:11)

(西德:9)

(西德:10)

By finding the closest reference vector of each solution in population POP, the solu-
tions are grouped into different subobjective spaces. 然而, the number of solutions

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Evolutionary Computation Volume 29, 数字 4

A Self-Adaptive Response Strategy for DMOPs

in some subobjective spaces may be more than one or may be zero. To ensure that each
subobjective space has at least one solution, we also find the closest solution for each
reference vector as follows:

Xi =

(西德:6)
X|(西德:2)(F (X), r i ) = max
1≤j ≤N

(西德:7)

(西德:9)

(西德:10)

(西德:2)(F (xj ), r i )

x ∈ POP, i = 1, 2, . . . . . . , 氮
(西德:2)(F (xj ), r i ) = r i × (F (xj ) − Z)时间
(西德:11)
(西德:11)
(西德:11) ×
(西德:11)r i
(F (xj ) − Z)

(西德:11)
(西德:11)

(西德:11)
(西德:11) ,

(5)

where Xi represents the corresponding closest solution of the reference vector r i.

The Maxi-Min Fitness Function

3.1.2
The maxi-min strategy (Luce and Raiffa, 2012; 罗尔斯, 2009) was first introduced into
multiobjective optimization in Balling (2003). Here we also adopt the maxi-min fitness
function to compute the fitness of the individual, which can account for both conver-
gence and diversity so that no additional measures for diversity is needed.

Supposing that an MOEA of a size N is used to solve an m-objective multiobjective
k is the k − th objective function value of the i − th individual.
minimization problem, f i
然后, the maxi-min fitness function of i − th individual of the population is denoted
作为:

fitnessi = maxj (西德:5)=i,j =1,…,氮 (mink=1,…,米(f i
k

− f j

k )), i = 1, 2, . . . , 氮.

(6)

We can see from Eq. (6) that if the maxi-min fitness of an individual is greater
比 0, this individual is regarded as a dominated individual, while if the maxi-min
fitness of an individual is less than 0, this individual is regarded as a nondominated
个人. If the maxi-min fitness of an individual is 0, this individual is regarded
as a weakly dominated individual. The maxi-min fitness function is well suited for
multiobjective optimization, because it can penalize individuals in a crowded region
while rewarding those in a less crowded region (Balling, 2003).

Environmental Selection

3.1.3
Before environmental selection, parent population POPiter is merged with the offspring
population Qiter to generate a combined population PQiter . According to Eq. (6), 这
fitness values of all individuals in the population PQiter are calculated. The individual
with the fitness value being less than 0 is regarded a nondominated individual. 如果
number of nondominated individuals in PQiter is greater than N, all nondominated in-
dividuals are retained. If the number of nondominated individuals in PQiter is less than
or equal to N, then the individuals in PQiter are sorted by the maxi-min fitness in an
ascending order, and remaining the first N individuals. The final operation is to find
the closest individual of each reference vector, which forms the new parent population
with N individuals. The main purpose of this selection strategy is to allow the individ-
uals that dominate others and contribute more to the maintenance of the population
diversity to have a larger chance to survive.

3.1.4 Update Strategy
During each generation, each reference vector should find its closest solution, and if one
of the following conditions is satisfied, the existing solution in the subobjective space
will be replaced by the newly generated solution: (1) the new solution dominates the
existing one; (2) the two solutions do not dominate each other, but the new one is closer
to the reference vector of the related subobjective space. The first condition makes sure

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that a nondominated solution is preserved so that the population converges gradually
to the Pareto front. 相比之下, the second condition aims to maintain the diversity of
人口.

3.2

Environmental Change Detection

The most common way to detect environmental changes is to re-evaluate a portion of
existing solutions (Carlisle and Dozier, 2000). This is realized by randomly choosing a
number of individuals from the current population and re-evaluating them, then check
the differences of the objective values or the constraint conditions of the chosen individ-
uals between two generations. If there is a significant difference, then an environmental
change is considered to have happened.

在这项工作中, the following similarity detection operator presented in Farina et al.

(2004) and Liu et al. (2017) is used to detect environmental changes:
(西德:11)
(西德:11)
(西德:11)
(西德:11) ,

F (xj , iter ) − F (xj , iter + 1)
F (xj , iter )max − F (xj , iter )min

ε(iter ) = 1
nε

(西德:11)
(西德:11)
(西德:11)
(西德:11)

nε(西德:8)

j =1

(7)

where nε is the number of the individuals chosen from the current population. 如果
ε(iter ) > ˜ε (˜ε is the threshold value, 0.00001), an environmental change occurs.

3.3

Self-Adaptive Response Strategy

This section mainly introduces the ideas of SRS in detail. SRS integrates several widely
used response strategies and continuously determines the probability of selecting the
response strategies according to their contributions to the performance improvement in
the previous environments, thereby adaptively selecting the best response strategy for
different problems. 这边走, the self-adaptive response strategy can be used to handle
unknown environmental changes.

In our empirical studies, SRS integrates five commonly used response strate-
吉斯, namely RDI (Deb et al., 2007), MDI (Deb et al., 2007), LPS (Zhou et al., 2007),
FPS (Hatzakis and Wallace, 2006), and PPS (Zhou et al., 2014). The description of
the each of these is shown in Section S1 of the Supplementary material, 可以在
https://doi.org/10.1162/evco_a_00289. In the following, we will present the details of
SRS.

Proposed SRS

3.3.1
It should be pointed out that according to the original reference, the prediction strat-
egy in PPS and FPS comes into play only after 23 environmental changes. 因此, SRS
adopts RDI as the response strategy in the first 23 environmental changes so that the
five strategies can fairly compete with each other, because the original LPS, FPS, 和
PPS all use the RDI strategy when the prediction model is not yet ready to work in the
early environmental changes.

When the 24th environmental change is detected, SRS selects the five different re-
sponse strategies for initializing the new population at an equal probability of 0.2. 那
is to say, the five different response strategies all generate a new population, then SRS
selects 20% of its individuals from each population to form a new population. By assign-
ing a specific label to the individuals generated by each of the five response strategies,
SRS is able to track them during the evolution. Suppose RDI is labeled as 1, MDI, LPS,
FPS, and PPS are labeled as 2, 3, 4, 和 5, 分别. 1, 2, 3, 4, 和 5 are considered as
the meta labels (MLs). 最后, the individuals generated by RDI, MDI, LPS, FPS,
and PPS are assigned labels as (1,1,1), (2,2,2), (3,3,3), (4,4,4), 和 (5,5,5), 分别.

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数字 1: Self-adaptive Response Strategy (SRS).

We will explain at the end of this section why we assign label (1,1,1) (IE。, the label of an
individual consists of three MLs) 而不是 (1) to an individual. In the entire evolution-
ary process, each individual will always carry a triplet label (我, J, K ) to trace the origin
of the individual, where I, J, K ∈ {1, 2, 3, 4, 5}.

When a new (except for the first 24) environmental change is detected, SRS can
obtain a set of nondominated solutions, each having a label in the format of (我, J, K ).
Then it examines how much performance improvement each individual in the previous
environment has contributed to the performance improvement by counting the total
number of MLs corresponding to each response strategy. 例如, 的数量
the MLs in one individual’s label (1,1,1) 是 3. Then we calculate the contribution ratio
of each response strategy. Assume we have 100 individuals in total, the number of all
the MLs is 300, and the number of RDI’s label (1) 是 45. Thus the contribution ratio of
strategy RDI is 0.15. A strategy having a higher contribution ratio is assigned a greater
probability for generating new individuals to respond to new environmental changes.
Once an environmental change occurs, the five response strategies all generate a new
人口, then SRS selects different ratios of individuals from each population to form
a new population and passes them to the new environment. This self-adaptive response
strategy is illustrated in Figure 1.

When an environmental change is detected, SRS will respond to the change and
generate a new initial population for the new environment. In the new environment,
we use MOEA-OSD to search for nondominated solutions starting from the new initial
人口. 所以, it is important to assign labels in the format of (我, J, K ) to these
individuals generated by the crossover operator and mutation operator, 因为

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桌子 1: Determining the label of an offspring after the crossover in Situation 1.

RDI

MDI

LPS

FPS

PPS

机器学习
NL
p
应用程序 (Rp)
Dp

1
2
2/9
2/3 (1)
−1/3

2
3
3/9
1 (1)
0

3
1
1/9
1/3 (0)
1/3

4
2
2/9
2/3 (1)
−1/3

5
1
1/9
1/3 (0)
1/3

label assignment process determines whether the contribution of each response strategy
is correctly calculated.

3.3.2 Determining the Label of an Offspring Individual
When a mutation operator is applied, the label of the offspring individual is the same as
the label of its parent individual. 例如, if the label of a parent individual is (1,3,2),
the label of the offspring individual is (1,3,2), 也. When a differential crossover operator
is applied to generate offspring, which uses three parent individuals to generate an
offspring individual, the label of the offspring individual is difficult to determine.

Algorithm 2 describes the main components of determining the label of an offspring
individual generated by the differential crossover operator. We provide the following
three instances to elaborate how Algorithm 2 作品.

A) Situation 1 (Lines 6–8 in Algorithm 2). The following gives a simple example:
suppose the labels of the three parent individuals are (1,2,3), (4,2,4), 和 (1,2,5), 重新指定-
主动地. Now we count the number of different MLs in these three individuals’ labels,
which is called as the number of labels (NL). 因此, the number of RDI’s label (1) 是 2,
resulting in a probability of 2/9 for the response strategy RDI, this probability is de-
noted as p. 然后, we compute the probability of each ML appearing in the label of the
offspring individual generated by the crossover operator, which is denoted as appear-
ing probability (应用程序). Ap after being rounded is expressed as a rounded probability (Rp).
The difference between Ap and Rp is denoted as difference probability (Dp = Ap − Rp),
which could also be considered as a redundancy probability. For RDI, its label (1) 将要
appear in the label of offspring individual with a probability of 3*2/9 = 2/3. This will
= 1. 因此, based on Rp, we can determine the final label
be rounded to 1; 那是, Rp1
of the offspring individual. 在这个例子中, the label of the offspring individual will be
(1,2,4). The detailed steps for computing the labels of the offspring individual are listed
表中 1.

乙) Situation 2 (Lines 10–18 in Algorithm 2). Here is a simple example: suppose the
labels of three parent individuals are (1,1,2), (2,3,4), 和 (3,5,4), 分别. As shown
表中 2, according to Rp, there are four “1”s. Thus the label of the offspring individual
will be (1,2,3,4), which is no longer a triplet (我, J, K ). 然后, according to Dp, the response
strategies corresponding to labels 1, 2, 3, 4 will have a negative redundancy (−1/3).
所以, we will randomly select three labels from 1, 2, 3, 和 4 to obtain the label of
the offspring individual; 那是, the offspring individual in this situation will be labeled
作为 (1,2,3) 或者 (1,2,4) 或者 (1,3,4) 或者 (2,3,4).

C) Situation 3 (Lines 20–23 in Algorithm 2). The following is a simple instance: sup-
pose the labels of the three parent individuals are (1,1,2), (1,2,3), 和 (2,2,5). As shown
表中 3, according to Rp, we can see that only part of the label of the offspring indi-
vidual can be determined, 这是 (1,2), since there are only two ”1”s. 根据

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桌子 2: Determining the label of an offspring after crossover in Situation 2.

RDI

MDI

LPS

FPS

PPS

机器学习
NL
p
应用程序 (Rp)
Dp

1
2
2/9
2/3 (1)
−1/3

2
2
2/9
2/3 (1)
−1/3

3
2
2/9
2/3 (1)
−1/3

4
2
2/9
2/3 (1)
−1/3

5
1
1/9
1/3 (0)
1/3

桌子 3: Determining the label of an offspring after crossover in Situation 3.

RDI MDI

LPS

FPS

PPS

机器学习
NL
p
应用程序 (Rp)
Dp

1
3
3/9
1 (1)
0

2
4
4/9
4/3 (1)
1/3

3
1
1/9
1/3 (0)
1/3

4
0
0
0 (0)
0

5
1
1/9
1/3 (0)
1/3

Dp, the response strategies corresponding to the labels 2, 3, 和 5 will have a positive
冗余 1/3, while response strategies of the label 1 和 4 have no redundancy.
Thus we can randomly select a label from 2, 3, 和 5 to assign the entire label of the off-
spring individual. 因此, the offspring individual may be labelled as (1,2,2), (1,2,3),
或者 (1,2,5).

总之, in order to use SRS to determine the probability of selecting different
response strategies and to generate the new initial population, it is important to deter-
mine the individuals’ labels in triplet during the evolution process of MOEA-OSD.

The reason why we assign a triplet label instead of a single number to an individual
is as follows. Assume the individuals generated by RDI, MDI, LPS, FPS, and PPS are
assigned a label of 1, 2, 3, 4, 和 5, 分别. Now let us assume that the labels of
the three parent individuals are 1, 4, 和 5, and then we need to randomly select a
label from 1, 4, 和 5 to assign a label to the offspring individual, 那是, (1) 或者 (4) 或者
(5). If the label of the offspring individual is randomly selected as 1, only RDI will be
selected, FPS and PPS may be ignored outright, while RDI, FPS, and PPS are equal in this
situation. 所以, we assign a label (1,1,1) to the individual instead of (1) to increase
the accuracy of selecting different response strategies.

3.4 Overall MOEA-OSD/SRS

A flowchart of the overall MOEA-OSD/SRS is given in Figure 2. Firstly, an initial
population of size N is randomly generated (POP0), and a set of N reference vectors
r 1, r 2, . . . , r N evenly distributed in the whole objective space are generated. 然后, 这
algorithm identifies the closest solution of each reference vector and initializes the ref-
erence point Z.

At the beginning of each generation, the similarity detection operator is applied to
detect environmental changes. If a change is detected, SRS is triggered to select the best
response strategy. 否则, MOEA-OSD continues to evolve using the given strategy.
In the process, the assignment of the individual’s labels is very important, as presented
in Section 3.3.

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A Self-Adaptive Response Strategy for DMOPs

数字 2: The flowchart of the proposed MOEA-OSD/SRS.

The above process is repeated until the termination criterion is met. When an envi-

ronmental change is detected, the PF of the previous environment is outputted.

3.5 Computational Complexity

For MOEA-OSD/SRS, in each generation, computational resources are mainly con-
sumed by MOEA-OSD.

As described in Algorithm 1, MOEA-OSD consists of the following components at
each generation: finding the closest solution of each reference vector, performing genetic
运营, evaluating the maxi-min fitness value, performing environmental selection
and the update strategy. The time complexity for finding the closest solution of each ref-
erence vector is O(m × N 2), where m is the number of objectives and N is the population
尺寸. The computational complexity for the genetic operations is O(m × N ). The calcu-
lation of the maxi-min fitness value holds a computational complexity of O(m × N 2).
此外, the computational complexity for environmental selection and the update
strategy are O(m × N 2) 和O(m × N ), 分别.

总结一下, the total computational complexity of MOEA-OSD/SRS for one gen-

eration is O(m × N 2).

4

Empirical Studies

This section verifies the performance of MOEA-OSD/SRS by conducting comprehen-
sive empirical studies, including a comparison of SRS with other six effective response
策略, a comparison of MOEA-OSD/SRS with other seven state-of-the-art dynamic
multiobjective optimization algorithms, and a comparison of MOEA-OSD with other
six state-of-the-art multiobjective optimization algorithms.

4.1

Benchmark Functions

Eleven benchmark functions F1-F11 are used to test the performance of MOEA-
OSD/SRS. F1, F2, and F3 are DMOP1, DMOP2, and DMOP3 of the DMOP suite (Goh
and Tan, 2009A), F4, F5, F6, F7, and F8 are FDA1, FDA2, FDA3, FDA4, and FDA5 of

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the FDA suite (Farina et al., 2004), F9 and F10 are JY2 and JY5 of the JY suite (Jiang and
哪个, 2017A), and F11 (Zhou et al., 2014) is a function with noncyclic changes. The defini-
tions and characteristics of those functions are given in Section S2 of the Supplementary
材料.

The changing dynamics of these benchmark functions is implemented by Eq. (8):
(西德:12)

(西德:13)

t = 1
nT

t
τT

,

(8)

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where nT and τT is the severity and frequency of the environmental changes, 重新指定-
主动地.

To indicate the performance of DMOAs in different types of environmental changes,
we define the following settings of (τT , nT ), 包括 (10,10), (15,10), 和 (20,10). 为了
each benchmark function, the environment changes 100 times in each run and each
algorithm is performed 20 independent runs on each benchmark function.

4.2

Performance Indicators

在这项工作中, we adopt the following performance indicators to assess the convergence
and diversity performance of the solution sets obtained by the algorithms under com-
parison, which are Inverted generational distance (IGD) (刘等人。, 2017; Zhou et al.,
2007), Spacing (Schott, 1995), and Hypervolume (Zitzler and Thiele, 1999). The descrip-
tion of the three performance indicators is presented in Section S3 of the Supplementary
材料.

4.3 Comparison between SRS and Other Six Effective Response Strategies

To assess the effectiveness of the proposed SRS, here we compared the performance of
SRS with six most widely used response strategies on 11 benchmark functions.

4.3.1 Comparison Algorithms and Parameter Settings
This section verifies the better performance of SRS by conducting a comparison
with six well-known response strategies, that are RDI, MDI, LPS, FPS, and PPS,
and self-adaptive diversity introduction (SADI; 刘, Zheng et al., 2014). For com-
parisons, each of these response strategies is embedded into MOEA-OSD, 这是
denoted as MOEA-OSD/SRS, MOEA-OSD/RDI, MOEA-OSD/MDI, MOEA-OSD/
LPS, MOEA-OSD/FPS, MOEA-OSD/PPS, and MOEA-OSD/SADI, 分别. 自从
SRS integrates five response strategies, 那是, RDI, MDI, LPS, FPS, and PPS, we there-
fore compare SRS with the five strategies. We also compare SRS with SADI, as SADI is
a self-adaptive response strategy, 也. SADI adaptively determines the diversity intro-
duction ratio according to the severity of environmental changes, which is described
in Section S1 of the Supplementary material in detail. Most parameter settings of each
strategy are based on the recommendations in the original references and the details are
presented in Section S4.1 of the Supplementary material.

Experimental Results

4.3.2
桌子 4 presents the statistical results of I GD of the solution sets obtained by the seven
algorithms averaged over 20 runs; S and H V metric values can be found in the Section
S4.2 of Supplementary material. 表中 4, the best results of the first five algorithms
using a different single response strategy are highlighted. These results indicate the
best single response strategy among RDI, MDI, LPS, FPS, and PPS for each benchmark

504

Evolutionary Computation Volume 29, 数字 4

A Self-Adaptive Response Strategy for DMOPs

桌子 4: The statistical results of I GD for different algorithms. “†”, “§”, and “≈” denote
the performance of MOEA-OSD/SRS is better than, worse than, or comparable with that
of the best of the first five algorithms.

功能

(τT , nT )

Statistic

F1

(20,10) 意思是

Std

(15,10) 意思是

Std

(10,10) 意思是

Std

F2

(20,10) 意思是

Std

(15,10) 意思是

Std

(10,10) 意思是

Std

F3

(20,10) 意思是

Std

(15,10) 意思是

Std

(10,10) 意思是

Std

F4

(20,10) 意思是

Std

(15,10) 意思是

Std

(10,10) 意思是

Std

F5

(20,10) 意思是

Std

(15,10) 意思是

Std

(10,10) 意思是

Std

F6

(20,10) 意思是

Std

(15,10) 意思是

Std

(10,10) 意思是

Std

F7

(20,10) 意思是

Std

(15,10) 意思是

Std

(10,10) 意思是

Std

F8

(20,10) 意思是

Std

(15,10) 意思是

Std

(10,10) 意思是

Std

F9

(20,10) 意思是

Std

(15,10) 意思是

Std

(10,10) 意思是

Std

MOEA MOEA MOEA MOEA MOEA MOEA MOEA
-OSD
-OSD
-OSD
/SRS
/FPS
/RDI

-OSD
/SADI

-OSD
/MDI

-OSD
/LPS

-OSD
/PPS

0.0044
5.71乙-04
0.0054
8.45乙-04
0.0089
0.0016

0.0082
1.08乙-04
0.0118
1.54乙-04
0.0228
3.79乙-04

0.0064
6.62乙-05
0.0086
1.75乙-04
0.0148
4.22乙-04

0.0073
8.01乙-05
0.0099
1.18乙-04
0.0180
4.18乙-04

0.0060
6.86乙-05
0.0072
1.88 乙-04
0.0100
2.03乙-04

0.0131
8.68乙-04
0.0186
0.0012
0.0273
9.16乙-04

0.0577
2.32乙-04
0.0625
2.93乙-04
0.0732
7.28乙-04

0.0435
0.0011
0.0518
9.43乙-04
0.0678
0.0024

0.0094
1.28乙-04
0.0133
2.21乙-04
0.0245
8.78乙-04

0.0043
2.17乙-04
0.0049
4.05乙-04
0.0080
0.0020

0.0077
7.07乙-04
0.0107
1.51乙-04
0.0192
7.51乙-04

0.0071
6.81乙-05
0.0085
1.43乙-04
0.0150
6.64乙-04

0.0069
7.48乙-05
0.0091
1.05乙-04
0.0151
3.57乙-04

0.0058
1.40乙-04
0.0066
1.01乙-04
0.0087
2.05乙-04

0.0156
0.0016
0.0206
0.0014
0.0320
0.0025

0.0581
2.27乙-04
0.0633
4.18乙-04
0.0763
0.0011

0.0446
0.0011
0.0539
0.0015
0.0781
0.0017

0.0086
7.63乙-04
0.0177
1.40乙-04
0.0198
3.66乙-04

0.0060
9.62乙-05
0.0073
2.49乙-04
0.0156
0.0023

0.0056
4.78乙-05
0.0067
7.85乙-05
0.0099
2.26乙-04

0.0050
3.81乙-05
0.0059
7.21乙-05
0.0082
1.09乙-04

0.0054
3.35乙-05
0.0063
5.12乙-05
0.0087
1.59乙-04

0.0073
1.65乙-04
0.0088
1.81乙-04
0.0130
2.98乙-04

0.0084
4.69乙-04
0.0108
8.34乙-04
0.0159
8.79乙-04

0.0541
2.28乙-04
0.0571
2.45乙-04
0.0630
5.21乙-04

0.0386
3.66乙-04
0.0448
8.88乙-04
0.0548
0.0019

0.0062
4.58乙-05
0.0074
5.86乙-05
0.0103
1.85乙-04

0.0043
4.83乙-04
0.0051
6.58乙-04
0.0079
0.0014

0.0071
1.23乙-04
0.0096
2.31乙-04
0.0169
7.60乙-04

0.0060
7.46乙-05
0.0079
1.47乙-04
0.0127
5.16乙-04

0.0065
9.65乙-05
0.0084
1.84乙-04
0.0140
4.71乙-04

0.0059
9.86乙-05
0.0068
1.65乙-04
0.0088
1.51乙-04

0.0128
0.0016
0.0169
0.0020
0.0242
0.0018

0.0569
4.09乙-04
0.0612
3.86乙-04
0.0715
0.0010

0.0422
0.0012
0.0491
0.0010
0.0654
0.0024

0.0053
0.0012
0.0061
3.88乙-04
0.0111
0.0024

0.0061
1.01乙-04
0.0079
1.57乙-04
0.0133
4.40乙-04

0.0058
5.80乙-05
0.0073
1.44乙-04
0.0126
5.69乙-04

0.0057
8.06乙-05
0.0070
1.68乙-04
0.0108
3.39乙-04

0.0065
1.40乙-04
0.0079
3.76乙-04
0.0121
0.0015

0.0109
9.33乙-04
0.0160
0.0017
0.0271
0.0034

0.0575
4.00乙-04
0.0628
8.25乙-04
0.0804
0.0033

0.0441
0.0012
0.0544
0.0018
0.0874
0.0085

0.0149
0.0035
0.0255
0.0056
0.0507
0.0147

0.0088
1.40乙-04
0.0139
3.60乙-04
0.0297
0.0016

0.0064
1.23乙-04
0.0085
1.00乙-04
0.0142
4.64乙-04

0.0088
1.40乙-04
0.0113
2.22乙-03
0.0226
0.0011

0.0073
6.93乙-04
0.0095
7.43乙-04
0.0198
0.0036

0.0117
0.0011
0.0213
0.0029
0.0301
0.0017

0.0630
3.15乙-04
0.0676
3.82乙-04
0.0730
0.0018

0.0838
0.0012
0.0863
0.0014
0.1001
0.0030

0.0042
3.49乙-05
0.0047
5.51乙-05
0.0072
1.30乙-04

0.0056
3.27乙-05
0.0067
5.58乙-05
0.0098
1.35乙-04

0.0050
2.99乙-05
0.0059
5.49乙-05
0.0081
1.71乙-04

0.0054
3.98乙-05
0.0062
1.44乙-04
0.0086
2.09乙-04

0.0057
7.39乙-05
0.0066
1.23乙-04
0.0085
1.54乙-04

0.0083
7.04乙-04
0.0109
0.0011
0.0160
0.0010

0.0468
0.0011
0.0508
0.0020
0.0572
0.0040

0.0372
0.0025
0.0440
0.0026
0.0538
0.0054

0.0075
1.70乙-04
0.0102
3.49乙-04
0.0170
5.04乙-04

0.0064
1.08乙-04
0.0079
1.83乙-04
0.0123
3.27乙-04

0.0109
4.79乙-04
0.0166
5.95乙-04
0.0324
0.0027

0.0061
7.86乙-05
0.0071
5.76乙-05
0.0097
3.33乙-04

†/
§/
p-value ≈
≈

0.0947

0.6761

0.6791

0.5309

0.6761

0.2446

0.5006

0.1984

0.1258

0.2446

0.0235

0.0982

0.0122

0.0367

0.5815

0.5815

0.2979

0.2979

0.0122

0.0027

0.0235

0.0963

0.1542

0.0583

0.0216

0.0122

0.0216

≈

≈

≈

≈

≈

≈

≈

≈

≈

†

≈

†

†

≈

≈

≈

≈

†

†

†

≈

≈

≈

†

†

†

Evolutionary Computation Volume 29, 数字 4

505

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右. 刘等人.

功能

(τT , nT )

Statistic

桌子 4: Continued.

MOEA MOEA MOEA MOEA MOEA MOEA MOEA
-OSD
-OSD
-OSD
/SRS
/FPS
/RDI

-OSD
/SADI

-OSD
/MDI

-OSD
/LPS

-OSD
/PPS

F10

(20,10)

(15,10)

(10,10)

F11

(20,10)

(15,10)

(10,10)

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

0.0062
2.79乙-05
0.0065
3.05乙-05
0.0073
9.04乙-05

0.0060
1.73乙-05
0.0062
2.79乙-05
0.0067
5.76乙-05

0.2911
0.0217
0.3552
0.0227
0.5452
0.0178

0.4092
0.0661
0.7125
0.0088
0.9708
0.1062

0.0068
2.18乙-05
0.0076
6.39乙-05
0.0094
1.31乙-04

0.1174
5.41乙-04
0.1666
0.0015
0.2780
0.0249

0.0060
1.73乙-05
0.0062
2.52乙-05
0.0066
9.42乙-05

0.0062
4.93乙-05
0.0064
7.65乙-05
0.0072
2.10乙-04

0.0103
4.48乙-04
0.0154
0.0010
0.0286
0.0030

0.0060
3.24乙-05
0.0062
3.80乙-05
0.0067
3.01乙-04

0.1654
0.0147
0.2651
0.0368
0.5454
0.0057

0.1393
0.0435
0.3119
0.1604
0.6613
0.3908

0.0555
0.0354
0.4015
0.0133
0.6536
0.0180

0.1156
0.0033
0.1632
0.0038
0.2785
0.0116

†/
§/
p-value ≈
≈

0.8345

0.0601

0.1437

0.0122

0.2453

0.6985

≈

≈

†

≈

≈

function in the presence of different environmental changes (columns 4 到 8). The experi-
mental results (I GD) of the proposed algorithm and the best of the first five algorithms
are also analyzed using the Wilcoxon rank-sum test (Derrac et al., 2011). 表中 4,
p-value represents the probability of two set of I GD obtained by two different algo-
rithms come from the same distribution, “†”, “§”, and “≈” denote the performance of
the proposed algorithm is better than, worse than, or comparable to that of the best of
the first five algorithms, 分别.

表中 4, 柱子 10 provides the results of MOEA-OSD/SRS in terms of I GD for
different settings, which are all as good as or better than the best results obtained by
the five algorithms. From the statistical results of Wilcoxon rank-sum test, we can see
that on these 11 benchmark functions with three environmental settings (33 instances
in total), the proposed SRS significantly outperforms the best of the first five compared
response strategies on 11 instances and performs equally well on the rest 22 instances.
All the above results confirm that SRS can indeed select the most suited response
strategy for different benchmark functions, thereby being capable of solving DMOPs
subject to unknown environmental changes.

此外, the 9th column of Table 4 also presents the results of I GD obtained by
MOEA-OSD/SADI. Based on these results, we can conclude that MOEA-OSD/SRS has
a better performance than MOEA-OSD/SADI on all instances.

数字 3 presents the change of the probability of selecting different response strate-
gies on F1 and F2 when (τT , nT ) is set to (10,10). We can see that the final selected re-
sponse strategy matches the experimental results in Table 4; 那是, MDI is the best
response strategy for F1 because the PS of F1 is fixed, and LPS is the best for F2 in that
the PS of F2 changes with time.

此外, the average running time of 20 independent runs of all algorithms on
all benchmark functions when (τT , nT ) is set to (10,10), is listed in Table 5 of the Supple-
mentary material. 平均而言, MOEA-OSD/SRS is a little bit time-consuming. 这是
reasonable since SRS needs to implement all five response strategies. When an environ-
mental change occurs, the five response strategies all generate a new population, 然后
SRS selects different ratios of individuals from each population to form a new popula-
tion to respond to the environmental change. It should be pointed out that the increase
in computational complexity might be negligible compared to the very time-consuming
fitness evaluations in many expensive real-world optimization problems (Jin and Send-
hoff, 2002; Jin et al., 2018).

506

Evolutionary Computation Volume 29, 数字 4

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A Self-Adaptive Response Strategy for DMOPs

桌子 5: The statistical results of I GD for different algorithms. “†”, “§”, and “≈” denote
the performance of MOEA-OSD/SRS is better than, worse than, and comparable with
that of the corresponding comparison algorithm.

功能

(τT , nT )

Statistic

-II-A

-II-B

-MEDA -MEDA

SGEA MOEA/D

DNSGA DNSGA

FPS
-RM

PPS
-RM

F1

F2

F3

F4

F5

F6

F7

F8

F9

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

0.0158 †
0.0075
0.0238 †
0.0088
0.0360 †
0.0125

0.0113 †
5.83乙-04
0.0168 †
5.69乙-04
0.0368 †
0.0024

0.0099 †
7.23乙-04
0.0158 †
0.0010
0.0508 †
0.0031

0.0099 †
3.05乙-04
0.0137 †
5.41乙-04
0.0299 †
0.0026

0.1547 †
8.89乙-05
0.1553 †
2.65乙-04
0.1565 †
6.42乙-04

0.1436 †
0.0043
0.1516 †
0.0090
0.1713 †
0.0594

0.1372 †
0.0122
0.2255 †
0.0227
0.4007 †
0.0348

0.2609 †
0.0074
0.2797 †
0.0073
0.3504 †
0.0143

0.0124 †
0.0064
0.0194 †
0.0078
0.0307 †
0.0097

0.0111 †
2.76乙-04
0.0169 †
5.78乙-04
0.0381 †
0.0035

0.0096 †
5.75乙-05
0.0163 †
0.0011
0.0542 †
0.0064

0.0096 †
2.54乙-05
0.0141 †
9.29乙-04
0.0297 †
0.0019

0.1548 †
2.90乙-04
0.1552 †
1.77乙-04
0.1564 †
7.24乙-04

0.1435 †
0.0041
0.1461 †
0.0039
0.1872 †
0.0707

0.1585 †
0.0091
0.2424 †
0.0218
0.4180 †
0.0324

0.2595 †
0.0078
0.2833 †
0.0080
0.3566 †
0.0145

0.0153 †
6.25乙-04
0.0257 †
0.0012
0.0607 †
0.0048

0.0156 †
7.24乙-04
0.0255 †
0.0013
0.0625 †
0.0054

0.009 5 †
0.0025
0.0128 †
0.0020
0.0244 †
0.0064

0.0109 †
3.00乙-05
0.0197 †
9.71乙-04
0.0430 †
0.0021

0.0084 †
2.92乙-04
0.0116 †
4.06乙-04
0.0210 †
0.0010

0.0084 †
1.60乙-04
0.0129 †
4.97乙-04
0.0266 †
0.0012

0.0071 †
1.54乙-04
0.0079 †
2.20乙-04
0.0111 †
3.88乙-04

0.0110 †
5.57乙-04
0.0114 †
4.53乙-04
0.0199 †
0.0011

0.0994 †
0.0012
0.1033 †
0.0017
0.1138 †
0.0016

0.0998 †
0.0043
0.0948 †
0.0017
0.0840 †
0.0016

0.0468 †
1.28乙-04
0.0471 †
2.01乙-04
0.0514 †
6.13乙-04

0.0095 †
0.0025
0.0127 †
0.0025
0.0253 †
0.0125

0.0093 †
6.03乙-04
0.0171 †
0.0020
0.0973 †
0.0444

0.0119 †
9.47乙-04
0.0229 †
0.0015
0.0655 †
0.0114

0.0119 †
4.02乙-04
0.0130 †
0.0014
0.0626 †
0.0233

0.0107 †
0.0012
0.0153 †
0.0014
0.0385 †
0.0084

0.0173 †
0.0020
0.0362 †
0.0075
0.1972 †
0.0582

0.0988 †
0.0016
0.1076 †
0.0020
0.1303 †
0.0067

0.0988 †
0.0016
0.1059 †
0.0073
0.1204 †
0.0093

0.0087 †
0.0022
0.0112 †
0.0028
0.0156 †
0.0039

0.0074 †
5.57乙-04
0.0096 †
8.70乙-04
0.0144 †
9.15乙-04

0.0718 †
0.0088
0.0869 †
0.0103
0.1303 †
0.0105

0.0718 †
7.63乙-04
0.0091 †
0.0012
0.0135 †
0.0012

0.0067 †
0.0011
0.0084 †
0.0028
0.0099 †
0.0014

0.0108 †
0.0042
0.0134 †
0.0033
0.0175 †
0.0051

0.0764 †
0.0012
0.0907 †
0.0018
0.1307 †
0.0058

0.0622 †
0.0044
0.0746 †
0.0045
0.0980 †
0.0079

0.0541 †
0.0155
0.0477 †
3.91乙-04
0.0667 †
0.0141

0.0077 †
1.76乙-04
0.0103 †
3.05乙-04
0.0158 †
4.33乙-04

0.0101 †
0.0040
0.0145 †
0.0028
0.0264 †
7.78乙-04

0.0098 †
1.31乙-04
0.0115 †
7.43乙-04
0.0236 †
0.0074

0.0088 †
1.63乙-04
0.0121 †
5.09乙-04
0.0256 †
0.0049

0.0088 †
5.42乙-04
0.0121 †
3.05乙-04
0.0263 †
0.0016

0.0087 †
8.47乙-05
0.0094 †
1.93乙-04
0.0138 †
0.0021

0.0153 †
2.84乙-04
0.0176 †
0.0012
0.0213 †
0.0046

0.0910 †
0.0076
0.1053 †
0.0021
0.1206 †
0.0034

0.0713 †
0.0018
0.0833 †
0.0042
0.1072 †
0.0044

0.0098 †
1.77乙-04
0.0114 †
0.0013
0.0159 †
0.0026

Evolutionary Computation Volume 29, 数字 4

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0.0094 †
0.0020
0.0120 †
0.0088
0.0220 †
0.0125

0.0089 †
6.42乙-04
0.0101 †
5.69乙-04
0.0153 †
0.0014

0.0085 †
6.16乙-04
0.0118 †
0.0020
0.0189 †
0.0018

0.0085 †
1.41乙-04
0.0108 †
5.86乙-04
0.0228 †
0.0026

0.0079 †
7.75乙-04
0.0088 †
8.95乙-04
0.0118 †
5.58乙-04

0.0144 †
0.0010
0.0153 †
0.0040
0.0195 †
0.0018

0.0795 †
0.0012
0.0948 †
0.0020
0.1181 †
0.0040

0.0680 †
0.0017
0.0785 †
0.0043
0.1001 †
0.0041

0.0095 †
4.55乙-04
0.0110 †
0.0010
0.0168 †
0.0030

MOEA
-OSD
/SRS

0.0042
3.49乙-05
0.0047
5.51乙-05
0.0072
1.30乙-04

0.0056
3.27乙-05
0.0067
5.58乙-05
0.0098
1.35乙-04

0.0050
2.99乙-05
0.0059
5.49乙-05
0.0081
1.71乙-04

0.0050
3.98乙-05
0.0062
1.44乙-04
0.0086
2.09乙-04

0.0057
7.39乙-05
0.0066
1.23乙-04
0.0085
1.54乙-04

0.0083
7.04乙-04
0.0109
0.0011
0.0160
0.0010

0.0468
0.0011
0.0508
0.0020
0.0572
0.0040

0.0372
0.0025
0.0440
0.0026
0.0538
0.0054

0.0061
7.86乙-05
0.0071
5.76乙-05
0.0097
3.33乙-04

507

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右. 刘等人.

桌子 5: Continued.
FPS
-RM

PPS
-RM

DNSGA DNSGA

功能

(τT , nT )

Statistic

-II-A

-II-B

-MEDA -MEDA

F10

(20,10)

(15,10)

(10,10)

F11

(20,10)

(15,10)

(10,10)

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

4.45乙-04

0.0067 ≈ 0.0066 ≈ 0.0058 ≈ 0.0058 ≈ 0.0042 §
5.62乙-04
1.25乙-05
3.14乙-05
0.0060 ≈ 0.0062 ≈ 0.0060 ≈ 0.0062 ≈ 0.0043 §
2.35乙-05
3.82乙-04
3.17乙-04
8.53乙-05
4.25乙-05
0.0044 ≈
0.0060 ≈ 0.0067 ≈ 0.0073 †
0.0059 §
5.45乙-05
4.03乙-04
1.10乙-04
2.91乙-04
3.60乙-04

5.51乙-05

SGEA MOEA/D
0.0063 ≈
9.70乙-05
0.0068 ≈
2.89乙-04
0.0087 †
6.01乙-04

0.4826 †
0.0127
0.5971 †
0.0269
0.9034 †
0.1505

0.5418 †
0.0242
0.5576 †
0.0331
0.8868 †
0.0053

0.2260 †
0.0035
0.3376 †
0.0046
0.7904 †
0.0215

0.2560 †
0.0053
0.3988 †
0.0101
0.8852 †
0.0232

0.1241 †
0.0019
0.1889 †
0.0154
0.4915 †
0.0718

0.8585 †
0.0160
0.8673 †
0.1085
0.8484 †
0.0975

Immune
GDE3
0.0060 ≈
3.85乙-04
0.0065 ≈
4.28乙-04
0.0070 ≈
5.55乙-04

0.3746 †
0.0252
0.2150 †
0.0146
0.5976 †
0.0127

MOEA
-OSD
/SRS

0.0060
3.24乙-05
0.0062
3.80乙-05
0.0067
3.01乙-04

0.1156
0.0033
0.1632
0.0038
0.2785
0.0116

4.4 Comparison between MOEA-OSD/SRS and the Other Seven DMOAs

To evaluate the performances of the entire MOEA-OSD/SRS for dynamic multiobjective
优化, we compare it with seven popular DMOAs.

4.4.1 Compared Algorithms and Parameter Settings
To further demonstrate the effectiveness of MOEA-OSD/SRS, we compare it with the
other seven state-of-the-art DMOAs, which are DNSGA-II-A (Deb et al., 2007), DNSGA-
II-B (Deb et al., 2007), RM-MEDA based on FPS (denoted as FPS-RM-MEDA) (周
等人。, 2014), RM-MEDA based on PPS (denoted as PPS-RM-MEDA) (Zhou et al., 2014),
SGEA (Jiang and Yang, 2017乙), and multiobjective evolutionary algorithm based on de-
作品 (MOEA/D) (Zhang and Li, 2007), and Immune Generalized Differential
进化 3 (Immune-GDE3) (Martínez-Peñaloza and Mezura-Montes, 2018). The de-
tailed parameter settings can be found in Section S5.1 of the Supplementary material.

Experimental Results of I GD, S, and H V

4.4.2
Tables 5–7 list the statistical results of I GD, S, and H V obtained by MOEA-OSD/SRS
and seven compared algorithms over 20 runs when (τT , nT ) is set as (10,10), (15,10),
和 (20,10). In those tables, the results of the best performing algorithm, 那是, 这
one obtaining the smallest mean values of I GD, the smallest mean values of S, 和
largest values of H V , are highlighted. The Wilcoxon rank-sum test (Derrac et al., 2011) 是
conducted to compare the significance of difference between MOEA-OSD/SRS and the
seven compared algorithms. Since it is necessary to compare the performance difference
between MOEA-OSD/SRS and each of comparison algorithm, in order to better present
the experimental results, we just use symbols “†”, “§”, and “≈” to express the statistical
analysis results. “†”, “§”, and “≈” indicate that the performance of MOEA-OSD/SRS
is better than, worse than, or comparable to that of the algorithm under comparison,
分别.

同时, it can be seen from Tables 5–7, among the three different types of en-
vironmental changes, 什么时候 (τT , nT ) is set as (20,10), the performance of different al-
gorithms is better than the situations that (τT , nT ) is set as (10,10) 和 (15,10), 和
performance of the algorithm is getting better and better with the increasing of τT .

A) From Table 5 (I GD): we observe that MOEA-OSD/SRS outperforms other al-
gorithms on most benchmark functions, except on F10, while SGEA performs signif-
icantly better than all compared algorithms on F10. For F10, as seen from Table 4,

508

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A Self-Adaptive Response Strategy for DMOPs

桌子 6: The statistical results of S for different algorithms. “†”, “§”, and “≈” denote the
performance of MOEA-OSD/SRS is better than, worse than, and comparable with that
of the corresponding comparison algorithm.

功能

(τT , nT )

Statistic

-II-A

-II-B

-MEDA -MEDA

SGEA MOEA/D

DNSGA DNSGA

FPS
-RM

PPS
-RM

F1

F2

F3

F4

F5

F6

F7

F8

F9

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

0.0089 †
0.0017
0.0145 †
0.0045
0.0210 †
0.0053

0.0073 †
1.28乙-04
0.0098 †
3.21乙-04
0.0168 †
9.60乙-04

0.0068 §
1.66乙-04
0.008 §
1.35乙-04
0.0164 †
6.98乙-04

0.0067 §
1.30乙-04
0.0083 §
2.87乙-04
0.0137 †
7.39乙-04

0.0066 §
1.17乙-04
0.0066 §
1.76乙-04
0.0069 §
3.54乙-04

0.0296 †
0.0079
0.0229 †
0.0133
0.0383 †
0.0117

0.0667 †
0.0014
0.0801 †
0.0015
0.0989 †
0.0040

0.1008 †
0.0013
0.1056 †
0.0020
0.1188 †
0.0021

0.0114 †
6.02乙-04
0.0157 †
0.0012
0.0271 †
0.0014

0.0089 †
0.0017
0.0123 †
0.0044
0.0206 †
0.0099

0.0073 †
1.15乙-04
0.0098 †
2.23乙-04
0.0166 †
7.80乙-04

0.0067 §
1.07乙-05
0.0088 §
1.69乙-04
0.0166 †
6.11乙-04

0.0067 §
1.03乙-04
0.0083 §
2.19乙-04
0.0137 †
8.04乙-04

0.0066 §
7.90乙-05
0.0066 §
2.02乙-04
0.0068 §
2.81乙-04

0.0305 †
0.0081
0.0319 †
0.0071
0.0350 †
0.0136

0.0686 †
0.0018
0.0794 †
0.0029
0.0982 †
0.0029

0.1002 †
0.0012
0.1076 †
0.0014
0.1181 †
0.0014

0.0116 †
5.46乙-04
0.0153 †
7.15乙-04
0.0273 †
0.0012

0.0109 †
0.0151
0.0145 †
0.0371
0.0195 †
0.0385

0.0222 †
0.0089
0.0437 †
0.0133
0.0919 †
0.0161

0.0155 †
0.0016
0.0155 †
0.0050
0.0234 †
0.0051

0.0118 †
0.0018
0.0387 †
0.0128
0.0650 †
0.0211

0.0086 §
0.0017
0.0089 §
0.0016
0.0137 †
0.0034

0.0184 †
0.0028
0.0300 †
0.0081
0.0647 †
0.0154

0.1118 †
0.0124
0.1239 †
0.0095
0.1408 †
0.0069

0.1537 †
0.0066
0.1239 †
0.0095
0.1869 †
0.0071

0.0177 †
0.0058
0.0315 †
0.0058
0.0640 †
0.0077

0.0119 †
0.0039
0.0196 †
0.0103
0.0171 †
0.0144

0.0077 †
1.97乙-04
0.0123 †
0.0011
0.0243 †
0.0043

0.0059 †
1.70乙-04
0.0077 †
1.15乙-04
0.0085 †
5.62乙-04

0.0073 †
1.79乙-04
0.0086 †
2.49乙-04
0.0093 †
1.91乙-04

0.0060 §
0.0079 §
0.0010
4.30乙-04
0.0111 ≈ 0.0062 §
5.28乙-04
0.0022
0.0075 §
0.0153 †
4.79乙-04
0.0011

0.0074 §
0.0012
0.0092 §
8.80乙-04
0.0236 †
0.0085

0.0064 §
2.92乙-04
0.0068 §
1.89乙-04
0.0100 §
5.88乙-04

0.0167 †
0.0027
0.0288 †
0.0087
0.0626 †
0.0144

0.0705 †
0.0016
0.0837 †
0.0079
0.0975 †
0.0075

0.0050 §
8.94乙-05
0.0062 §
2.48乙-04
0.0089 §
3.86乙-04

0.0053 §
2.26乙-04
0.0058 §
1.98乙-04
0.0071 §
2.44乙-04

0.0456 †
0.0011
0.0472 †
7.51乙-04
0.0542 †
0.0013

0.0332 §
6.52乙-04
0.0398 §
0.0020
0.0529 §
0.0016

0.0436 §
0.0804 §
0.0044
6.05乙-04
0.0470 §
0.1403 †
0.0052
6.20乙-04
0.0550 §
0.1539 †
0.0068
0.0015
0.0086 ≈ 0.0064 §
1.44乙-04
7.97乙-04
0.0084 §
0.0131 †
2.70乙-04
0.0012
0.0124 †
0.0268 †
3.47乙-04
0.0056

0.0080 †
1.59乙-04
0.0087 †
8.46乙-04
0.0099 †
1.85乙-04

0.0085 †
1.06乙-04
0.0101 †
1.21乙-04
0.0115 †
5.23乙-04

0.0115 †
8.81乙-05
0.0158 †
1.61乙-04
0.0145 †
3.60乙-04

0.0118 †
8.83乙-05
0.0135 †
4.52乙-04
0.0154 †
1.92乙-03

0.0123 †
2.04乙-04
0.0138 †
2.28乙-04
0.0155 †
1.91乙-04

0.0432 †
0.0075
0.0577 †
0.0019
0.0690 †
0.0019

0.0925 †
0.0012
0.0927 †
1.99乙-04
0.0930 †
6.19乙-04

0.1451 †
0.0012
0.1488 †
0.0015
0.1551 †
0.0016

0.0188 †
1.74乙-04
0.0320 †
2.05乙-04
0.0642 †
2.38乙-04

Evolutionary Computation Volume 29, 数字 4

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GDE3

0.0075 †
2.60乙-04
0.0082 †
0.0088
0.0095 †
0.0125

0.0080 †
5.83乙-04
0.0092 †
5.69乙-04
0.0100 †
4.58乙-04

0.0099 §
7.23乙-04
0.0088 §
0.0010
0.0080 §
0.0031

0.0065 §
3.05乙-04
0.0085 §
5.41乙-04
0.0120 †
0.0026

0.0084 §
8.89乙-04
0.0088 §
2.65乙-04
0.0089 §
6.42乙-04

0.0285 †
0.0043
0.0299 †
0.0090
0.0305 †
0.0059

0.0825 †
0.0012
0.0766 †
0.0022
0.0959 †
0.0034

0.0899 §
0.0074
0.0988 †
0.0073
0.1095 †
0.0143

0.0099 †
6.25乙-04
0.0115 †
0.0012
0.0138 †
0.0048

MOEA
-OSD
/SRS

0.0058
9.94乙-05
0.0064
1.14乙-04
0.0075
1.20乙-04

0.0069
5.88乙-05
0.0076
4.92乙-05
0.0090
7.43乙-05

0.0108
7.99乙-05
0.0112
5.01乙-05
0.0116
4.72乙-05

0.0109
6.80乙-05
0.0112
1.50乙-04
0.0117
8.76乙-05

0.0113
7.24乙-05
0.0115
3.70乙-05
0.0118
5.33乙-05

0.0128
2.03乙-04
0.0131
7.09乙-05
0.0142
3.70乙-04

0.0553
0.0016
0.0586
0.0020
0.0617
7.90乙-04

0.0999
0.0024
0.1035
0.0022
0.1083
0.0027

0.0085
5.75乙-05
0.0091
4.07乙-05
0.0099
4.95乙-05

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桌子 6: Continued.
FPS
-RM

PPS
-RM

DNSGA DNSGA

功能

(τT , nT )

Statistic

-II-A

-II-B

-MEDA -MEDA

SGEA MOEA/D

F10

(20,10)

(15,10)

(10,10)

F11

(20,10)

(15,10)

(10,10)

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

0.0113 ≈ 0.0117 †
0.0040
0.0026
0.0157 †
0.0142 †
0.0031
0.0025
0.0207 †
0.0213 †
0.0042
0.0049

0.0828 †
0.0088
0.1054 †
0.0132
0.2448 †
0.0168

0.0801 †
0.0092
0.1566 †
0.0225
0.2996 †
0.0261

0.0090 §
0.0015
0.0130 †
0.0034
0.0221 †
0.0052

0.6646 †
0.0572
0.7423 †
0.0656
0.7673 †
0.0427

0.0065 §
1.48乙-04
0.0069 §
4.19乙-04
0.0082 §
8.03乙-04

0.4047 †
0.0326
0.5610 †
0.0329
0.5560 †
0.0440

0.0034 §
6.68乙-05
0.0037 §
5.36乙-04
0.0039 §
5.53E-4

0.0649 †
0.0207
0.0907 †
0.0134
0.1623 †
0.0238

0.0128 †
1.27乙-04
0.0158 †
3.01乙-05
0.0209 †
2.23乙-04

0.0885 †
0.0060
0.1083 †
0.0043
0.2480 †
0.0279

Immune
GDE3

0.0098 §
5.62乙-04
0.0106 §
3.17乙-04
0.0125 ≈
3.60乙-04

0.0778 †
0.0019
0.1022 †
0.0102
0.2470 †
0.0062

MOEA
-OSD
/SRS

0.0108
6.55乙-05
0.0111
9.43乙-05
0.0118
1.03乙-04

0.0633
0.0032
0.0807
0.0113
0.1323
0.0155

MOEA-OSD/SRS is able to select the most effective response strategy for F10, 所以
poor performance of MOEA-OSD/SRS on F10 may be attributed to the poor perfor-
mance of the static algorithm MOEA-OSD on F10.

乙) From Table 6 (S): it is noted that MOEA-OSD/SRS obtains the best results on F1,
F2, F6, and F11, while SGEA performs the best on other seven benchmark functions.
It implies that MOEA-OSD/SRS only maintains better distribution on few benchmark
功能, and MOEA-OSD/SRS is inferior to SGEA. From these experimental results,
we can conclude that MOEA-OSD/SRS should incorporate some strategy to improve
the diversity when dealing with DMOPs.

C) From Table 7 (H V ): we can see that the H V values obtained by the compared
algorithms are largely consistent with the I GD values presented in Table 7. 清楚地,
MOEA-OSD/SRS is more promising than the compared algorithms for solving DMOPs,
although it is outperformed by SGEA on F10. The possible reason for this has been
discussed previously in the analysis of I GD.

最后但并非最不重要的, recall that I GD and H V measure the performance of a solution
set in terms of both convergence and distribution. 因此, if an algorithm achieves
solutions with a poor performance in diversity, it is possible that it exhibits a good I GD
and H V value on account of a superior performance in convergence.

同时, MOEA-OSD/SRS is relatively better than the seven compared algo-
rithms on F11, which is a nonperiodic function, although all the algorithms are not able
to obtain better performance in terms of I GD, S, and H V .

4.4.3 Comparison of the Tracking Ability of the Eight Algorithms
It is necessary to investigate how an algorithm performs after changes and in different
environments after a change (Helbig and Engelbrecht, 2013A). 数字 4 provides the IGD
over time averaged over 20 runs when (τT , nT ) is set as (10,10). It can be clearly seen
that MOEA-OSD/SRS responds to the changes more stably and recovers faster on most
of the benchmark functions than the compared algorithms, thereby obtaining better
convergence performance. The only exception occurs on F10, where SGEA performs the
最好的, the I GD value obtained by MOEA-OSD/SRS fluctuates strongly, probably due to
poor convergence and diversity when a change occurs.

In addition to the above experimental results, the comparison results of the ob-
tained PF of the eight compared algorithms are also presented in Section S5.2 of the
Supplementary material. 同时, the average runtime of 20 independent runs of the

510

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A Self-Adaptive Response Strategy for DMOPs

桌子 7: The statistical results of H V for different algorithms. “†”, “§”, and “≈” denote
the performance of MOEA-OSD/SRS is better than, worse than, and comparable with
that of the corresponding comparison algorithm.

功能

(τT , nT )

Statistic

-II-A

-II-B

-MEDA -MEDA

SGEA MOEA/D

DNSGA DNSGA

FPS
-RM

PPS
-RM

F1

F2

F3

F4

F5

F6

F7

F8

F9

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

(20,10)

(15,10)

(10,10)

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

0.6436 †
0.0047
0.6411 †
0.0023
0.6368 †
0.0039

0.6404 †
3.19乙-04
0.6303 †
7.78乙-04
0.5962 †
0.0041

0.8597 †
9.54乙-04
0.8491 †
9.74乙-04
0.7949 †
0.0042

0.8607 †
2.11乙-04
0.8516 †
0.0011
0.8216 †
0.0054

0.7030 †
1.32乙-04
0.7005 †
2.63乙-04
0.6966 †
1.08乙-04

1.0147 †
0.0038
1.0010 †
0.0126
0.9988 †
0.0083

0.5294 †
0.0214
0.4084 †
0.0026
0.2902 †
0.0057

3.0936 †
0.0642
2.8396 †
0.0019
2.4662 †
0.0089

0.6405 †
0.0017
0.6403 †
0.0029
0.6330 †
0.0024

0.6403 †
7.82乙-04
0.6299 †
6.27乙-04
0.5888 †
0.0121

0.8586 †
4.75乙-05
0.8491 †
0.0025
0.7904 †
0.0045

0.8605 †
5.38乙-05
0.8526 †
2.45乙-04
0.8250 †
0.0053

0.7029 †
1.39乙-05
0.7011 †
1.02乙-04
0.6964 †
1.77乙-04

1.0154 †
0.0018
1.0060 †
0.0116
0.9985 †
0.0038

0.5201 †
0.0102
0.4119 †
0.0228
0.3036 †
0.0033

3.0475 †
0.0467
2.8047 †
0.0532
2.4445 †
0.0874

0.6891 †
9.54乙-04
0.6730 †
0.0014
0.6117 †
0.0022

0.6878 †
1.36乙-04
0.6689 †
0.0054
0.6055 †
0.0081

0.6456 †
8.76乙-04
0.6429 †
6.85乙-04
0.6356 †
0.0015

0.6398 †
0.0010
0.6231 †
0.0013
0.5800 †
0.0011

0.8611 †
3.82乙-04
0.8557 †
1.61乙-04
0.8392 †
0.0012

0.8617 †
1.94乙-04
0.8531 †
9.69乙-04
0.8313 †
0.0021

0.6598 †
2.55乙-04
0.6579 †
6.92乙-04
0.6529 †
2.91乙-04

1.0170 †
0.0017
1.0040 †
0.0020
0.9934 †
0.0117

0.5886 †
0.0021
0.5776 †
3.53乙-04
0.5571 †
0.0065

2.8990 †
0.0174
2.7445 †
0.0517
2.3414 †
0.0213

0.6937 †
2.02乙-04
0.6819 †
7.05乙-04
0.6508 †
0.0035

0.6448 †
0.0015
0.6439 †
1.59乙-04
0.6321 †
0.0018

0.6424 †
2.97乙-04
0.6254 †
0.0032
0.5894 †
0.0270

0.8542 †
0.0012
0.8383 †
0.0031
0.7627 †
0.0020

0.8632 †
1.64乙-04
0.8492 †
0.0020
0.8295 †
0.0011

0.6516 †
0.0011
0.6459 †
7.59乙-04
0.6037 †
0.0020

0.9985 †
0.0055
0.9923 †
0.0107
0.9857 †
0.0518

0.5821 †
0.0026
0.5642 †
0.0018
0.5007 †
0.0070

2.7988 †
0.0165
2.4886 †
0.1279
2.0190 †
0.0126

0.6979 †
2.72乙-04
0.6904 †
3.47乙-04
0.6307 †
0.0082

0.6488 †
2.35乙-04
0.6446 †
6.11乙-04
0.6435 †
0.0029

0.6468 †
0.0010
0.6426 †
0.0016
0.6340 †
8.67乙-04

0.7645 †
0.0106
0.7312 †
0.0045
0.6759 †
0.0034

0.8646 †
1.23乙-04
0.8611 †
1.63乙-04
0.8539 †
8.65乙-04

0.7034 †
8.67乙-04
0.7006 †
0.0028
0.6986 †
9.37乙-04

1.0169 †
0.0079
1.0105 †
0.0034
1.0015 †
0.0079

0.6724 †
9.74乙-04
0.6524 †
0.0027
0.6416 †
0.0010

3.4270 †
0.0044
3.3664 †
0.0023
3.2432 †
0.0084

0.6999 †
1.09乙-04
0.6964 †
1.75乙-04
0.6893 †
2.69乙-04

0.6434 †
5.69乙-04
0.6391 †
0.0030
0.6356 †
1.04乙-04

0.6417 †
1.62乙-04
0.6248 †
0.0011
0.5846 †
0.0079

0.7694 †
1.02乙-04
0.7606 †
8.70乙-04
0.7502 †
1.56乙-04

0.8602 †
2.04乙-04
0.8529 †
1.41乙-04
0.8301 †
0.0016

0.6256 †
1.76乙-05
0.6239 †
6.34乙-04
0.6175 †
0.0024

0.9788 †
9.14乙-04
0.9773 †
0.0049
0.9758 †
0.0048

0.5977 †
3.31乙-04
0.5887 †
2.22乙-04
0.5690 †
5.84乙-04

3.2993 †
0.0013
3.2760 †
0.0011
3.2101 †
0.0053

0.6874 †
8.49乙-04
0.6246 †
0.0013
0.6173 †
0.0041

Evolutionary Computation Volume 29, 数字 4

Immune
GDE3

0.6425 †
6.75乙-04
0.6399 †
0.0040
0.6365 †
0.0054

0.6406 †
5.83乙-04
0.6250 †
5.69乙-04
0.5885 †
0.0024

0.8588 †
7.23乙-04
0.8221 †
0.0010
0.7895 †
0.0031

0.8622 †
3.05乙-04
0.8541 †
5.41乙-04
0.8342 †
0.0026

0.6556 †
8.89乙-04
0.6531 †
2.65乙-04
0.6428 †
6.42乙-04

0.9988 †
0.0043
0.9956 †
0.0090
0.9915 †
0.0059

0.6022 †
0.0021
0.5999 †
0.0022
0.5901 †
0.0034

3.3345 †
0.0074
3.2985 †
0.0073
3.2564 †
0.0143

0.6957 †
6.25乙-04
0.6942 †
0.0012
0.6921 †
0.0048

MOEA
-OSD
/SRS

0.6550
6.69乙-05
0.6538
1.59乙-04
0.6500
3.63乙-04

0.6511
6.37乙-05
0.6488
1.43乙-04
0.6429
2.93乙-04

0.8682
7.26乙-05
0.8665
1.12乙-04
0.8619
1.89乙-04

0.8674
3.86乙-05
0.8658
3.37乙-04
0.8612
1.73乙-04

0.7043
1.00乙-04
0.7029
1.22乙-04
0.6997
1.28乙-04

1.0214
0.0056
1.0115
0.0026
1.0056
0.0010

0.7351
1.50乙-04
0.7220
0.0076
0.7166
2.40乙-04

3.5342
0.0019
3.4784
0.0023
3.4415
0.0475

0.7022
2.17乙-05
0.7011
2.72乙-04
0.7001
2.17乙-04

511

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右. 刘等人.

桌子 7: Continued.
FPS
-RM

PPS
-RM

DNSGA DNSGA

功能

(τT , nT )

Statistic

-II-A

-II-B

-MEDA -MEDA

F10

(20,10)

(15,10)

(10,10)

F11

(20,10)

(15,10)

(10,10)

意思是
Std
意思是
Std
意思是
Std

意思是
Std
意思是
Std
意思是
Std

0.7061 †
5.67乙-04
0.7043 †
1.38乙-04
0.7021 †
2.31乙-04

0.0734 †
0.0033
0.0639 †
0.0132
0.0900 †
9.56乙-04

0.7066 †
3.21乙-04
0.7041 †
3.77乙-04
0.7027 †
3.05乙-04

0.1444 †
0.0229
0.0943 †
0.0531
0.0413 †
0.0076

0.7057 †
1.18乙-05
0.7053 †
8.95乙-05
0.7034 †
4.33乙-04

0.4341 †
0.0017
0.3046 †
0.0130
0.0896 †
0.0136

0.7054 †
8.30乙-05
0.7046 †
5.22乙-04
0.7017 †
4.06乙-04

0.2560 †
0.0053
0.1155 †
0.0319
0.0445 †
0.0266

SGEA MOEA/D
0.7090 ≈
3.96乙-05
0.7089 ≈
2.08乙-05
0.7084 §
9.22乙-05

0.7064 †
1.42乙-04
0.7048 †
4.39乙-04
0.7032 †
3.29乙-04

0.5887 †
0.0023
0.5166 †
0.0122
0.2651 †
0.0365

0.3369 †
0.0116
0.3217 †
0.0243
0.2673 †
0.0133

Immune
GDE3

0.7068 †
5.62乙-04
0.7049 †
3.17乙-04
0.7038 †
3.60乙-04

0.4823 †
0.0127
0.4092 †
0.0099
0.1993 †
0.0085

MOEA
-OSD
/SRS

0.7086
2.17乙-05
0.7082
1.64乙-04
0.7064
0.0013

0.6009
9.01乙-04
0.5463
0.0054
0.4142
4.68乙-04

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数字 3: The change of probability of selecting different response strategies.

compared algorithms are presented in Section S5.3 of the Supplementary material, 版本-
ifying that MOEA-OSD/SRS is the fastest algorithm among the eight algorithms.

此外, it should be noted that the baseline multiobjective optimization al-
gorithm is also an important component of a DMOA, which heavily affects the per-
formance of DMOAs. 因此, we also make a comparison of MOEA-OSD with other six
well-known multiobjective optimization algorithms, indicating that MOEA-OSD is the
best-performing and computationally most efficient algorithm. The details are shown
in Section S6 of the Supplementary material.

最后但并非最不重要的, we also conduct a sensitivity analysis to investigate the influence
of some parameters on the performance of MOEA-OSD/SRS, including the severity of
the environmental changes, 那是, nT , the proportion of diversity introduction in RDI
and MDI, differential crossover probability (CR), differential scale factor (F), and Gaus-
sian mutation probability. The details are in Section S7 of the Supplementary material.

5

PID Controller Parameter Turning

To verify the performance of the proposed MOEA-OSD/SRS on real applications, 我们
use it to tune the coefficients of the PID controller of a dynamic system.

The PID controller is very widely used to obtain the control according to the de-
viation between the desired input value and the actual output value of the system. 在
事实, the parameters of the system may change due to the aging of the equipment or the

512

Evolutionary Computation Volume 29, 数字 4

A Self-Adaptive Response Strategy for DMOPs

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数字 4: The average IGD over 20 runs versus the time.

Evolutionary Computation Volume 29, 数字 4

513

右. 刘等人.

桌子 8: The statistical results of I GD, S, and H V for different algorithms.

DNSGA DNSGA

FPS
-RM

PPS
-RM

指标

Statistic

-II-A

-II-B

-MEDA -MEDA

SGEA MOEA/D

I GD

S

H V

意思是
Std
意思是
Std
意思是
Std

4.9289 †
0.0579
1.8673 †
0.3479
2.3793 †
0.0093

4.8479 †
0.1245
1.6659 †
0.5166
2.3823 †
0.0164

3.7097 †
0.1196
0.1303 †
0.1156
2.4038 †
0.0033

3.6284 †
0.0373
3.6284 †
0.0373
2.4050 †
0.0011

3.9999 †
0.0815
0.0417 §
0.0019
2.4005 †
0.0030

4.0811 †
0.1001
0.0639 †
0.0108
2.3891 †
0.0032