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IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 21, NO. 1, FEBRUARY 2017 65

A Steady-State and Generational Evolutionary

Algorithm for Dynamic Multiobjective

Optimization

Shouyong Jiang and Shengxiang Yang, Senior Member, IEEE

Abstract—This paper presents a new algorithm, called

steady-state and generational evolutionary algorithm, which

combines the fast and steadily tracking ability of steady-state

algorithms and good diversity preservation of generational algo-

rithms, for handling dynamic multiobjective optimization. Unlike

most existing approaches for dynamic multiobjective optimiza-

tion, the proposed algorithm detects environmental changes and

responds to them in a steady-state manner. If a change is detected,

it reuses a portion of outdated solutions with good distribution

and relocates a number of solutions close to the new Pareto front

based on the information collected from previous environments

and the new environment. This way, the algorithm can quickly

adapt to changing environments and thus is expected to provide a

good tracking ability. The proposed algorithm is tested on a num-

ber of bi- and three-objective benchmark problems with different

dynamic characteristics and difﬁculties. Experimental results

show that the proposed algorithm is very competitive for dynamic

multiobjective optimization in comparison with state-of-the-art

methods.

Index Terms—Change detection, change response, dynamic

multiobjective optimization, steady-state and generational evo-

lutionary algorithm.

I. INTRODUCTION

ANY real-world multiobjective optimization problems

(MOPs) are dynamic in nature, whose objective func-

tions, constraints, and/or parameters may change over time.

Due to the presence of dynamisms, dynamic MOPs (DMOPs)

pose big challenges to evolutionary algorithms (EAs) since

any environmental change may affect the objective vector,

constraints, and/or parameters. As a result, the Pareto-optimal

set (POS), which is a set of mathematical solutions to MOPs,

and/or the Pareto-optimal front (POF) that is the image of

POS in the objective space, may change over time. Then, the

Manuscript received November 13, 2015; revised March 7, 2016 and

May 3, 2016; accepted May 10, 2016. Date of publication August 1, 2016;

date of current version January 26, 2017. This work was supported by the

Engineering and Physical Sciences Research Council (EPSRC) of U.K. under

Grant EP/K001310/1 and the National Natural Science Foundation (NNSF)

of China under Grant 61673331. (Corresponding author: Shengxiang Yang.)

The authors are with the Centre for Computational Intelligence,

School of Computer Science and Informatics, De Montfort University,

Leicester, LE1 9BH, U.K. (e-mail: [email protected];

[email protected]).

This paper has supplementary downloadable material available at

http://ieeexplore.org, provided by the authors. This supplementary material

provides the formulation of the test problems used in the paper and some

supplementary experimental results. This material is 102 KB in size.

Color versions of one or more of the ﬁgures in this paper are available

online at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/TEVC.2016.2574621

optimization goal is to track the moving POF and/or POS and

obtain a sequence of approximations over time.

DMOPs can be deﬁned in different ways, according to the

nature of dynamisms [15], [41], [54]. In this paper, we mainly

consider the following kind of DMOPs:

min F(x, t) = ( f

(x, t),...,f

(x, t))

s.t.

⎧

⎪

⎨

⎪

⎩

(x, t) = 0, i = 1,...,n

(x, t) ≥ 0, i = 1,...,n

x ∈ 

, t ∈ 

(1)

where M is the number of objectives, n

and n

are the number

of equality and inequality constraints, respectively, 

⊆ R

is the decision space, t is the discrete time instance, 

⊆ R

is the time space, and F(x, t) : 

×

→ R

is the objective

function vector that evaluates solution x at time t.

In the past few years, there has been an increasing amount

of research interest in the ﬁeld of evolutionary multiob-

jective optimization as many real-world applications, like

thermal scheduling [42] and circular antenna design [3], have

at least two objectives that conﬂict with each other, i.e.,

they are MOPs. Due to multiobjectivity, the goal of solv-

ing MOPs is not to ﬁnd a single optimal solution but to

ﬁnd a set of tradeoff solutions. When an MOP involves

time-dependent components, it can be regarded as a DMOP.

Many real-life problems in nature are DMOPs, such as

planning [8], scheduling [12], [35], and control [15], [50].

There have been a number of contributions made to several

important aspects of this ﬁeld, including dynamism classi-

ﬁcation [15], [41], test problems [4], [15], [20], [23]–[26],

performance metrics [9], [15], [17]–[19], [41], [55], and

algorithm design [9], [12], [15], [18], [21], [28], [54], [55].

Among these, algorithm design is the most important issue

as it is the problem-solving tool for DMOPs.

Due to the presence of dynamisms, the design of a dynamic

multiobjective optimization EA (DMOEA) is different from

that of a multiobjective optimization EA (MOEA) for static

MOPs. Speciﬁcally, DMOEAs should not only have a fast

convergence performance (which is crucial to their tracking

ability), but also be able to address diversity loss whenever

there is an environmental change in order to explore the new

search space. Besides, if changes are not assumed to be know-

able, DMOEAs should be able to detect them in order not

to mislead the optimization process. This is because, when a

1089-778X

 2016 IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/

redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

66 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 21, NO. 1, FEBRUARY 2017

change occurs, the previously discovered POS may not remain

optimal for the new environment.

In principle, a change can be detected by re-evaluating

dedicated detectors [12], [18], [47], [54], [55] or assessing

algorithm behaviors [15], [32], [37]. The former is a easy-to-

use mechanism and allows “robust detection” [37]ifahigh

enough number of detectors is used, but it may require addi-

tional cost since detectors have to be re-evaluated at every

generation, and it may not be accurate when there is noise

in function evaluations. The latter does not need additional

function evaluations, but it may cause false positives and thus

make algorithms overreacting when no change occurs. Both

of them cannot guarantee that changes are detected [37].

On the other hand, whenever a change is detected, it is

often inefﬁcient to restart the optimization process from

scratch, although the restart strategy may be a good choice if

the environmental change is considerably severe [7]. In the

literature, various approaches have been proposed to handle

environmental changes, and they can be mainly categorized

into diversity-based approaches and convergence-based

approaches, according to their algorithm behaviors. Diversity-

based approaches focus on maintaining population diversity

whereas convergence-based ones aim to achieve a fast con-

vergence performance so that algorithms’ tracking ability are

guaranteed. Generally, population diversity can be handled by

increasing diversity using mutation of selected old solutions

or random generation of some new solutions upon the detec-

tion of environmental changes [12], [18], [55], maintaining

diversity throughout the optimization process [1], [2], [6], or

employing multipopulation schemes [18], [40]. Proper

diversity is helpful for exploring promising search

regions, but too much diversity may cause evolutionary

stagnation [5].

Convergence-based approaches try to exploit past infor-

mation for better tracking performance [7], especially when

the new POS is somewhat similar to the previous one or

environmental changes exhibit regular patterns. Accordingly,

recording relevant past information to be reused at a later stage

may be helpful for tracking the new POF as quickly as possi-

ble. The reuse of past information is closely related to the type

of environmental change and hence can be helpful for different

purposes [6]. If the environment changes periodically, relevant

information of the current POS can be stored in a memory

and can be directly reintroduced into the evolving population

when needed. This kind of strategy is often called memory-

based approaches and has been extensively studied in dynamic

multiobjective optimization [7], [8], [18], [22], [52]. In con-

trast, if the environment change follows a regular pattern, past

information can be collected and used to model the movement

of the changing POF/POS. Hence, the location of the new

POS can be predicted, helping the population quickly track

the moving POF. Prediction-based approaches have received

massive attention because most existing benchmark DMOPs

(e.g., the FDA test suite [15]) involve predictable charac-

teristics, and studies along this direction can be referred

to [

22], [28], [32], [33], [36], [47], [54], and [55].

Aside from the above-mentioned approaches, some stud-

ies concentrate on ﬁnding an insensitive robust POF instead

of closely tracking the moving POF [16], [27], [38].

Algorithm 1 Framework of SGEA

1: Input: N (population size)

2: Output: a series of approximated POFs

3: Create an initial parent population P

={x

,...,x

};

4: (A, P)

= EnvironmentSelection(P);

5: while stopping criterion not met do

6: for i

= 1toN do

7: if change detected and not responded then

8: ChangeResponse();

9: end if

10: y

= GenerateOffspring(P, A);

11: (P, A)

= UpdatePopulation(y);

12: end for

13: (A, P)

= EnvironmentSelection(P ∪ P);

14: Set P

= P;

15: end while

Robustness-based approaches assume that when the environ-

ment changes, the old obtained solution can still be used in

the new environment as long as its quality is acceptable [27].

However, the criterion for an acceptable optimal solution is

quite problem-speciﬁc, which may hinder the wide application

of these approaches.

Although a number of approaches have been proposed for

solving DMOPs, the development of DMOEAs is a rela-

tively young ﬁeld and more studies are greatly needed. In this

paper, a new algorithm, called steady-state and generational

EA (SGEA), is proposed for efﬁciently handling DMOPs.

SGEA makes most of the advantages of steady-state EAs in

dynamic environments [48] for environmental change detec-

tion and response. If a change is detected, SGEA reuses a

portion of old solutions with good diversity and exploits infor-

mation collected from both previous environments and the new

environment to relocate a part of its evolving population. At

the end of every generation, like conventional generational

EAs [13], [56], SGEA performs environmental selection to

preserve good individuals for the next generation. By mixing

the steady-state and generational manners, SGEA can adapt

to dynamic environments quickly whenever a change occurs,

providing very promising tracking ability for DMOPs.

The rest of this paper is organized as follows. Section II

describes the framework of the proposed SGEA, together with

detailed descriptions of each component of the algorithm.

Section III is devoted to presenting experimental settings for

comparison. Section IV provides experimental results and

comparison on tested algorithms. A further discussion of the

algorithm is offered in Section V. Section VI concludes this

paper with discussions on future work.

II. P

ROPOSED SGEA

The basic framework of the proposed SGEA is presented

in Algorithm 1. SGEA starts with an initial population P and

the initialization of an elitist population

P and an archive A

through environmental selection. In every generational cycle,

SGEA detects possible environmental changes and evolves the

population in a steady-state manner. For each population mem-

ber, if a change is detected, then a change response mechanism

is adopted to handle the detected change. After that, genetic

operation is applied to produce one offspring solution for the

population member, which is then used to update the parent

JIANG AND YANG: SGEA FOR DYNAMIC MULTIOBJECTIVE OPTIMIZATION 67

Algorithm 2 EnvironmentSelection(Q)

1: Input: Q (a set of solutions)

2: Output: A (archive), P (N elitists preserved)

3: Set A

=∅and P

=∅;

4: Assign a ﬁtness value to each member in Q;

5: for i

= 1to|Q| do

6: if F(i)<1 then

7: Copy x

from Q to A;

8: end if

9: end for

10: if |A| < N then

11: Copy the best N individuals in terms of their ﬁtness values

from Q to

12: else

13: if |A|==N then

14: Set P

= A;

15: else

16: Prune A to a set of N individuals by any truncation operator

and copy the truncated A to

17: end if

18: end if

population P and archive A. At the end of each generation,

P and

P are combined. Similar to generational EAs [13], [56]

or speciation techniques used in niching [5], [29], a genera-

tional environmental selection is conducted on the combined

population to preserve a population of good solutions for the

next generation. This way, SGEA can be regarded as a steady-

state and generational MOEA. In the following sections, the

implementation of each component of SGEA will be detailed

step by step.

A. Environmental Selection

The environmental selection procedure (Algorithm 2),

which aims to preserve a ﬁxed number of elitists from a

solution set Q after every generational cycle, starts with ﬁt-

ness assignment. Each individual i of Q is assigned a ﬁtness

value F(i), which is deﬁned as the number of individuals that

dominate [56] it, as follows:

F(i) =|{j ∈ Q|j ≺ i}| (2)

where |·| denotes the cardinality of a set and j ≺ i indi-

cates that j dominates i. It should be noted that, various

ﬁne-grained methods proposed in [14], [45], and [56] can

be used to assign ﬁtness values for individuals. However, the

ﬁtness assignment method used in this paper is relatively sim-

ple and computationally efﬁcient. Most importantly, when an

external individual e enters the set Q, the update of F(i) needs

only one dominance comparison between individuals e and i.

The easy-to-update property of this method will be clearly

embodied in the population update procedure (to be described

in Section II-C).

Afterwards, individuals having a ﬁtness value of zero are

identiﬁed as nondominated solutions and then copied to an

archive A.If|A| is smaller than the population size N

,the

best N individuals (including both dominated and nondomi-

nated ones) in terms of their ﬁtness values are preserved in

an elitist population

P. Otherwise, there can be two situations:

either the number of nondominated solutions ﬁts exactly the

population size, or there are too many nondominated solutions.

Algorithm 3 GenerateOffspring(P, A)

1: Input: P (parent population), A (archive population)

2: Output: y (offspring solution)

3: if rnd < 0.5 then

4: Perform binary tournament selection on P to select two distinct

individuals as the mating parents;

5: else

6: Randomly pick an individual from A and perform binary tour-

nament selection on P to select another distinct individual as

the mating parents.

7: end if

8: Apply genetic operators to generate a new solution y;

In the ﬁrst case, all nondominated solutions are copied to P.

In the second case, a truncation technique is needed to reduce

A to a population of N nondominated solutions such that the

truncated A have the best diversity possible. In SGEA, the kth

nearest neighbor truncation technique proposed in the strength

Pareto EA 2 (SPEA2) [56] is used to perform the truncation

operation, although we recognise there are other options, e.g.,

the farthest ﬁrst method [10], [11], which can also serve this

purpose. After that, solutions in the truncated A are copied

Note that, like classical generational MOEAs, such as the

nondominated sorting genetic algorithm II (NSGA-II) [13]

and SPEA2 [56], SGEA performs environmental selection at

the end of each generation. Thus, SGEA can be generally

categorized into generational MOEAs.

B. Mating Selection and Genetic Operators

Mating selection is an important operation before the pro-

duction of new offspring (line 10 of Algorithm 1). In this

paper, mating parents can be selected either from the parent

population P or the archive population A. The beneﬁt of such a

mating selection method has been extensively investigated on

static MOPs in a number of studies [30], [34], [44], [57]. While

selecting mating parents from P can maintain good population

diversity, selecting parents from A can signiﬁcantly improve

the convergence speed of the population, which is considerably

desirable in fast-changing environments. If a mating parent

is to be selected from P, SGEA performs a binary tourna-

ment selection according to individuals’ ﬁtness values. If not,

the mating parent can be randomly selected from the archive

population A.

Following the mating selection, genetic operators are

applied on the mating parents to generate a new offspring solu-

tion. In SGEA, the simulation binary crossover and polynomial

mutation are chosen as the recombination and mutation oper-

ators, respectively. The reproduction procedure is presented in

Algorithm 3.

C. Population Update

In SGEA, population update (line 11 of Algorithm 1)is

conducted on both the parent population P and archive popula-

tion A, which is detailed in Algorithm 4. The update operation

on P is in fact replacing the worst solution of P with the newly

generated solution y while the update on A is using y to update

the archived nondominated set. First, if y is not a duplicate

68 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 21, NO. 1, FEBRUARY 2017

Algorithm 4 UpdatePopulation(y)

1: Input: y (offspring solution)

2: Output: P (updated parent population), A (updated archive

population)

3: Set the ﬁtness value of y as zero: F(y)

= 0;

4: for i

= 1to|P| do

5: if y == x

then

6: Return;

7: end if

8: if y ≺ x

then

9: Add one to the ﬁtness value of x

: F(i)

= F(i) + 1;

10: end if

11: if y  x

then

12: Add one to the ﬁtness value of y: F(y)

= F(y) + 1;

13: end if

14: end for

15: Compute the individual in P having the highest ﬁtness value:

= i : argmax

{1≤i≤|P|}

F(i);

16: if F(y) ≤ F(

i) then

17: Set x

= y and F(

= F(y);

18: if F(y)<1 then

19: Remove all solutions in A that are dominated by y,andadd

y to A if A is not full;

20: end if

21: end if

solution, it will be compared with each member x

of P for

the dominance relation (lines 4–14 of Algorithm 4). If y domi-

nates x

(denoted as y ≺ x

), the ﬁtness value of x

is increased

by one. If y is dominated by x

(denoted as y  x

), the ﬁtness

value of y is increased by one. Then, the worst individual in

P with the highest ﬁtness value is identiﬁed, and if there are

two or more such individuals, a random one is selected. If y is

not worse than the identiﬁed individual x

in terms of the ﬁt-

ness value, the solution replacement takes place, as shown in

line 17 of Algorithm 4. Besides, if y is not dominated by any

member in P (which means its ﬁtness value is zero), it should

be further considered to update the archive population A if A

is not full. This means, the archive update occurs only when

y successfully enters the parent population. It can be observed

that, the ﬁtness assignment method used here is easy to update

an individual’s ﬁtness value, which helps SGEA conduct solu-

tion replacement in the parent population and archive update

in an efﬁcient manner.

D. Dynamism Handling

This section discusses two main aspects of dynamism han-

dling. One is change detection, a step to detect whether a

change has occurs during the evolutionary process. The other

is known as change response or change reaction, which takes

actions to quickly react to environmental changes so that the

population adapts to new environments rapidly.

1) Change Detection: Change detection can be

performed by either re-evaluating a portion of existing

solutions [12], [18], [47], [54], [55] or assessing some

statistical information of some selected population mem-

bers [15], [32], [37]. Since both methods choose a small

proportion of population members as detectors, detection

may fail if changes occur on nondetectors. On the contrary,

it will be computationally expensive if the whole population

members are chosen as detectors. Therefore, a good detection

method should strike a balance between the detection ability

and efﬁciency.

The proposed algorithm detects changes in a steady-state

manner, as shown in line 7 of Algorithm 1. In every gen-

eration, population members (in random order) are checked

one by one for discrepancy between their previous objective

values and re-evaluated ones. If a discrepancy exists in a pop-

ulation member, we assume a change is successfully detected

and there is no need to do further checks for the rest of popula-

tion members. When a change is detected, SGEA immediately

reacts to it in a steady-state manner. The detection method

is beneﬁcial to prompt and steady change reaction at the

cost of high computational cost. For efﬁciency, the number

of individuals re-evaluated for change detection is restricted

to a small percentage of the population size. It is worth not-

ing that, re-evaluation-based change detection methods assume

that there is no noise in function evaluations, i.e., they are not

robust. Thus, the proposed method may not be suitable for

detecting changes in noisy environments.

2) Change Response: If a change is successfully detected,

some actions should be taken to react to the environmental

change. A good change response mechanism must be able

to maintain a good level of population diversity and relocate

the population in promising areas that are close to the new

POS. Simply discarding old solutions and randomly reinitial-

izing the population is beneﬁcial to population diversity but

may be time-consuming for algorithms to converge. Likewise,

fully reusing old solutions for the new environment might

be misleading if the landscapes of two consecutive changes

are signiﬁcantly different. Also, this may cause the loss of

population diversity. As a consequence, algorithms may get

trapped into local minima or cannot ﬁnd all POF regions for

the new environment. For these reasons, in this paper the popu-

lation for the new environment consists of half of old solutions

and half of reinitialized solutions. The half old solutions

are selected by the farthest ﬁrst selection method [11], [43],

which was originally proposed to reduce an approximation

set to the maximum allowable size. The farthest ﬁrst selection

method has been reported to provide better approximation than

NSGA-II’s crowding distance [13] for unconstrained and con-

strained static MOPs [10], [11]. This method selects half of

old solutions that maximize the diversity in the objective space

(line 3 of Algorithm 5). The other half reinitialized solutions in

the new population are produced by a guess of the new location

of the changed POS. To make a correct or at least reasonable

guess, one must know two things, i.e., moving direction and

movement step-size. The following paragraphs contribute to

how to compute them.

Let C

be the centroid of POS and A

be the obtained

approximation set at time step t, then C

can be computed by



x∈A

x. (3)

The movement step-size S

to the new location of the

changed POS at time step t + 1 can be estimated by

=C

− C

t−1

 (4)

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