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"多目标优化问题档案导向均衡优化器" 本文主要讨论了多目标优化问题（MOPs）中的一种基于Epsilon支配的档案导向均衡优化器（Archive-Guided Equilibrium Optimizer）。这种优化器算法是基于均衡优化器（Equilibrium Optimizer，EO），其灵感来自于控制体积质量平衡模型，该模型使用粒子（解决方案）及其相应的体积和质量来搜索最优解决方案。多目标优化问题（MOPs）是指在实际应用中涉及到两个或多个相互冲突的目标函数，这些目标函数需要同时被优化。在解决这些问题时，需要一种有效的优化算法来寻找最优解决方案。本文提出的档案导向均衡优化器（GMOEO）算法正是为解决多目标优化问题而设计的。 GMOEO算法的主要思想是使用档案（Archive）来存储已知的最优解决方案，然后使用这些解决方案来指导搜索过程，找到更好的解决方案。同时，GMOEO算法还使用Epsilon支配来确保搜索的多样性和快速收敛性。在实际应用中，GMOEO算法可以应用于各种领域，例如机器学习、数据挖掘、控制系统等。该算法可以帮助研究人员和工程师更好地解决多目标优化问题，从而提高系统的性能和效率。此外，本文还讨论了GMOEO算法的实现细节，包括算法的伪代码、时间复杂度和空间复杂度的分析等。这些内容可以供研究人员和工程师参考和应用。本文提出的档案导向均衡优化器（GMOEO）算法是一种有效的多目标优化算法，能够解决实际应用中涉及到的多目标优化问题。该算法可以应用于各种领域，提高系统的性能和效率。本文的主要贡献包括： * 提出了基于Epsilon支配的档案导向均衡优化器（GMOEO）算法，该算法能够解决多目标优化问题。 * 讨论了GMOEO算法的实现细节，包括算法的伪代码、时间复杂度和空间复杂度的分析等。 * 介绍了GMOEO算法在实际应用中的应用前景，例如机器学习、数据挖掘、控制系统等。本文的贡献将有助于研究人员和工程师更好地解决多目标优化问题，从而提高系统的性能和效率。

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Citation: Chalabi, N.E.; Attia, A.;

Bouziane, A.; Hassaballah, M.;

Alanazi, A.; Binbusayyis, A. An

Archive-Guided Equilibrium

Optimizer Based on Epsilon

Dominance for Multi-Objective

Optimization Problems. Mathematics

2023, 11, 2680. https://doi.org/

10.3390/math11122680

Academic Editors: Siddhartha

Bhattacharya and Mohamed Abd

Elaziz

Received: 25 April 2023

Revised: 28 May 2023

Accepted: 5 June 2023

Published: 13 June 2023

Licensee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

Article

An Archive-Guided Equilibrium Optimizer Based on Epsilon

Dominance for Multi-Objective Optimization Problems

Nour Elhouda Chalabi

, Abdelouahab Attia

, Abderraouf Bouziane

, Mahmoud Hassaballah

3,4,

* ,

Abed Alanazi

and Adel Binbusayyis

Computer Science Department, Mohamed Boudiaf University of Msila, Msila 28000, Algeria

Computer Science Department, University Mohamed El Bachir El Ibrahimi of Bordj Bou Arreridj,

Bordj Bou Arreridj 34000, Algeria

Department of Computer Science, College of Computer Engineering and Sciences, Prince Sattam Bin

Abdulaziz University, AlKharj 16278, Saudi Arabia

Department of Computer Science, Faculty of Computers and Information, South Valley University,

Qena 83523, Egypt

Department of Software Engineering, College of Computer Engineering and Sciences, Prince Sattam Bin

Abdulaziz University, AlKharj 16278, Saudi Arabia

* Correspondence: mah.ali@psau.edu.sa

Abstract:

In real-world applications, many problems involve two or more conﬂicting objectives

that need to be optimized at the same time. These are called multi-objective optimization problems

(MOPs). To solve these problems, we introduced a guided multi-objective equilibrium optimizer

(GMOEO) algorithm based on the equilibrium optimizer (EO), which was inspired by control–volume–

mass balance models that use particles (solutions) and their respective concentrations (positions) as

search agents in the search space. The GMOEO algorithm involves the integration of an external

archive that acts as a guide and stores the optimal Pareto set during the exploration and exploitation

of the search space. The key candidate population also acted as a guide, and Pareto dominance was

employed to obtain the non-dominated solutions. The principal of

-dominance was employed to

update the archive solutions, such that they could then guide the particles to ensure better exploration

and diversity during the optimization process. Furthermore, we utilized the fast non-dominated

sort (FNS) and crowding distance methods for updating the position of the particles efﬁciently

in order to guarantee fast convergence in the direction of the Pareto optimal set and to maintain

diversity. The GMOEO algorithm obtained a set of solutions that achieved the best compromise

among the competing objectives. GMOEO was tested and validated against various benchmarks,

namely the ZDT and DTLZ test functions. Furthermore, a benchmarking study was conducted using

cone-

-dominance as an update strategy for the archive solutions. In addition, several well-known

multi-objective algorithms, such as the multi-objective particle-swarm optimization (MOPSO) and

the multi-objective grey-wolf optimization (MOGWO), were compared to the proposed algorithm.

The experimental results proved deﬁnitively that the proposed GMOEO algorithm is a powerful tool

for solving MOPs.

Keywords:

metaheuristic algorithms; multi-objective optimization; equilibrium optimizer; Pareto

solution set; e-dominance relation; cone-e-dominance

MSC: 68T27

1. Introduction

A number of real-life problems have typically been interpreted as optimization prob-

lems with multiple conﬂicting objectives [

] (e.g., water distribution networks (WDNs) [

the traveling salesman problem [

], and protein structures [

]). We are in an era where

such problems are increasing daily [

]. In addition, today’s decision-making problems

Mathematics 2023, 11, 2680. https://doi.org/10.3390/math11122680 https://www.mdpi.com/journal/mathematics

Mathematics 2023, 11, 2680 2 of 30

require us to consider large models of these NP-hard problems, both in terms of the number

of variables and constraints [

–

]. Therefore, these problems are handled and modeled

as multi-objective optimization problems (MOPs), where the goal is to ﬁnd the best set

of trade-off solutions—known as a Pareto optimal set or non-dominated solutions [

In other words, this type of optimization searches for acceptable compromises between

objectives—as compared to single-objective optimization, which is where only one solution

has to be found. Therefore, signiﬁcant attention has been given to this concept, and many

works have been proposed [

]. Meta-heuristics and evolutionary algorithms have been

widely adopted for solving some of the multi-objective optimization problems, including

non-dominating sort genetic algorithms (NSGAIIs) [

], where fast non-dominated sorting

was used. The extension of NSGAII, called NSGAIII [

], employed a non-dominated sort

and a reference point method. The PAES [

] and SPEA2 [

] employed an external archive

to store the non-dominated solutions; as such, algorithms have been quite successful and

are still used today [17,18].

MOPs have been the most common problems in several real-world applications [

Therefore, this ﬁeld has continued to evolve, thus ensuring that many other algorithms were

developed, such as the multi-objective evolutionary algorithm based on decomposition

(MOEAD) [

], where the problem is decomposed into a number of sub-problems and

each one is treated as a single-objective problem. Deb et al. [

] introduced the

-MOEA

algorithm, where the

-dominance relation was employed. Many other extensions of MOEA

have been proposed, including the uniform decomposition measurement (UMOEA/D) [

the MO-memetic algorithm (MOEA/D-SQA) [24], and many others [25,26].

In terms of meta-heuristics algorithms and, particularly, population-based algorithms,

algorithms that handle multi-objective problems (MOPs) have typically been an extension

of a single-objective-optimization algorithms but modeled in a way to solve MOPs. One

of the most well-known algorithms is the multi-objective particle swarm optimization

(MOPSO) method, which is based on the single-objective optimization algorithm type of

particle swarm optimization (PSO) [

]; it is a population-based algorithm inspired by

the biological behavior of birds in a ﬂock. The PSO has been proven to be a successful

algorithm that continues to be used for solving optimization problems [

]. Many extended

multi-objective versions of the PSO have been proposed. For instance, the swarm metaphor,

as proposed in [

], incorporated the Pareto dominance concept and crowding distances.

In a different work by

Cello et al. [30]

, another MOPSO was proposed that incorporated a

respiratory system to conserve the non-dominated solutions and to choose an instructor

that would guide the particles. The well-known algorithm of ant colony optimization

(ACO) and its

variants [31,32]

is another population-based algorithm. It was inspired by

ant behavior and designed to solve single-objective optimization problems. Furthermore, it

was improved to handle with MOPs accordingly, as in [33–35].

Following the same concept over the years, several other MOP algorithms were

developed by simply extending the single-objective version [

]. The cat swarm op-

timization (CSO) [

] method was extended by incorporating a Pareto ranking; thus, it

was then named the multi-objective cat-swarm optimization (MOCSO) [

]. The grey wolf

optimizer (GWO) [

], for example, was also extended by adding an external ﬁxed-size

archive, resulting in the multi-objective grey wolf optimization (MOGWO) [

] method.

Zouache et al. [

] introduced a guided multi-objective moth–ﬂame optimization (MOMFO)

method, which was an extension of the moth–ﬂame optimizer (MFO) [

]. In MFO, an

unlimited external archive was used to determine the non-dominated solutions, and the

fast non-dominated sort was adopted, along with crowding distances. Furthermore,

dominance was employed as an updated archive strategy. A more recent work attempted

to solve MOPs using EO by proposing a multi-objective equilibrium optimizer with an

exploration–exploitation dominance strategy (MOEO-EED) [44].

Recently, an equilibrium optimizer algorithm was presented to solve a single-objective

optimization problem [

]. The presented results of this algorithm showed that it was

able to outperform well-known algorithms. In this paper, based on the aforementioned

Mathematics 2023, 11, 2680 3 of 30

analysis and the extended versions, we present an extended version of EO called the

guided multi-objective equilibrium optimizer (GMOEO), which we used to solve MOPs.

The proposed extension employs an external archive through which to obtain the non-

dominated solutions and the crowding distances. In addition, it utilizes an exploration—

exploitation dominance, which is where the solutions update was controlled.Furthermore,

a Gaussian-based mutation strategy was suggested to boost the exploration and enhance

the exploitation. In this work, as compared with MOEO-EED, we attempted to approach

the concept using simple strategies.

The main contributions of this paper are summarized as follows.

• We propose a GMOEO method to solve multi-objective optimization problems;

•

We incorporated an external archive to store the non-dominated solutions and to

guide the particles toward the optimal Pareto set.

• e

-dominance was employed to update the archive solutions and to ensure improved

diversity, exploitation, and exploration. In addition, cone-

- dominance was employed

to update the archive solutions, and was compared with the e- dominance relation;

•

A fast non-dominating sort and crowding distances were introduced to preserve the

diversity and to ensure the convergence of the particles, as well as to ensure an efﬁcient

solution distribution;

•

The effectiveness of the proposed algorithm was validated through comprehensive

experiments conducted on different benchmarks, including ZDT and DTLZ test func-

tions. The performance was then compared with the known multi-objective optimiza-

tion algorithms.

The rest of the paper is organized as follows. Section 2 explains the basics of multi-

objective optimization problems, Pareto optimality, and EO. Section 3 introduces the

proposed GMOEO algorithm. The experimental results, comparisons, and discussion are

presented in Section 4. Finally, Section 5 contains the conclusions and suggestions for future

research directions.

2. Background Information

2.1. Multi-Objective Optimization Problems

Multi-objective optimization is the systematic process of simultaneously collecting and

optimizing conﬂicting objective functions. The optimization could involve minimization or

maximization, depending on the problem. A minimization problem could be formulated

as follows:

minimize

x∈<

(x), (i = 1, 2, . . . , M), (1)

subject to h

(x) = 0, j = (1, 2, 3, . . . , N), (2)

(x) ≤ 0, k = (1, 2, 3, . . . , L) (3)

where

(x)

, and

(x)

represent the decision functions of a vector

x = (x

. . . x

)

and

refers to decision variables. The variables

, and

refer to the objective

functions, search space, the numbers of inequality, and equality constraints, respectively. In

an MOP, it was difﬁcult to compare the obtained solutions with the relational arithmetic

operators. Therefore, the concept of Pareto optimal dominance provided a simple way

through which to compare solutions in the multi-objective search space. Furthermore, there

was a set of Pareto optimal solutions with a Pareto front image in the search space, rather

than a single optimal solution.

2.2. Pareto Dominance and Optimality

The main concept of the Pareto dominance relation comprised the following:

Mathematics 2023, 11, 2680 4 of 30

Deﬁnition 1

(

Pareto dominance

)

Let

and

be two solutions, where

dominates the other

solution s (denote as z < s ), iff:

∀i ∈

{

1, 2, 3, . . . , M

}

: f

(z) ≤ f

(s) and

∃j

{

1, 2, 3, . . . , M

}

: f

(z) < f

(s)

(4)

Solution z weakly dominated the other solution s (denoted as z 4 s ), iff:

∀i ∈

{

1, 2, 3, . . . , M

}

: f

(z) ≤ f

(s) (5)

Deﬁnition 2

(

A non-dominated set

)

The unique dominant solution where there is no other

solution that dominates them. Let

be a set of solutions, such that the non-dominated solutions are

members of the set D

⊆ D and are not dominated by any other solution in set D.

Deﬁnition 3. The Pareto solution is an optimal solution if it does not become dominated by other

solutions in the feasible search space.

6 ∃ s ∈ x| z ≺ s (6)

Deﬁnition 4 (Pareto optimal set). A set of all non-dominated solutions in the search space.

2.3. Equilibrium Optimizer (EO)

The equilibrium optimization algorithm [

] has recently been introduced to solve

optimization problems. The original innovation behind this algorithm was the control–

volume–mass balance models, where the mass-balance equation was employed for describ-

ing the concentration of a non-reactive constituent in the control volume. It is a ﬁrst-order

differential equation that could provide an understanding of the physics behind the con-

servation of mass as it enters and leaves the control volume. The solutions represent the

particles, while the concentrations represent the positions—similar to PSO. These particles

act as a search agent for reaching the optimal solution, which is the equilibrium state. Some

of EO’s key parameters that distinguish it from other algorithms include the following:

an equilibrium pool and a candidate; a generation rate; and updating process for the

particles’ positions.

The steps of the equilibrium optimizer are summarized in Algorithm 1. Similar to

other optimization algorithms, the EO began with the initialization of a random population.

As a unique approach, EO employed four of the best-so-far candidates

eq1

eq2

eq3

, and

eq4

as the equilibrium state, which were not initially known. These candidates were

necessary to determine the search pattern for the particles, to assist in the exploration of the

search space, and to participate in the update process. In addition to the four candidates,

an average candidate

ave

was calculated. The equilibrium pool was one of the key beneﬁts

of this optimizer, and the elements used in the construction of this vector were the ﬁve

candidates

eq.pool



eq1

, C

eq2

, C

eq3

, C

eq4

, C

ave



, which were chosen arbitrarily. This pool

participated in updating the concentration (position), where a random candidate would

be chosen. Other parameters, such as the generation-rate-control parameter

GCP

and the

generation probability

, were also employed. These two parameters contributed to

the construction of the generation rate

, which was another key parameter in the EO

algorithm.

GCP

was used to determine the potential contribution of

during the update

process, while

was used to determine which particle would update its status by using

the generation term.

Mathematics 2023, 11, 2680 5 of 30

Algorithm 1 Equilibrium optimizer (EO) [45]

Initialize the particles i = (1, 2, . .. , n)

Assign equilibrium candidates’ ﬁtness a large number

Assign the parameters a

, a

, GP = 0.5

while Iteration < Max

iteration

for i = 1: number of particles (n) do

Calculate ﬁtness of i

particle if ﬁtness (C

) < ﬁtness (C

eq1

) then

Replace C

eq1

withC

and ﬁtness (C

eq1

) with ﬁtness (C

)

else

if ﬁtness (C

) > ﬁtness (C

eq1

) and ﬁtness (C

) <ﬁtness (C

eq2

) then

Replace C

eq2

with C

and ﬁtness (C

eq2

) with ﬁtness (C

)

else

ﬁtness

) >

ﬁtness

eq1

)

and ﬁtness

) >

ﬁtness

eq2

)

and

) <

ﬁtness(C

eq3

) then

Replace C

eq3

with CC

and ﬁtness (C

eq3

) with ﬁtness (C

)

else

ﬁtness

) >

ﬁtness

eq1

)

and ﬁtness

) >

ﬁtness

eq2

)

and

)

>ﬁtness (C

eq3

) and (C

) < ﬁtness (C

eq4

) then

Replace C

eq4

with C

and ﬁtness (C

eq4

) with ﬁtness (C

)

end

ave

=(C

eq1

eq2

eq03

eq4

)/4

Equilibrium pool construction C

eq.pool

= C

eq1

eq2

eq3

eq4

ave

if iteration > 1 then

Accomplish memory saving

end

t = (1 −

iteration

Max

iteration

)

iteration

Max

iteration

)

for i = 1: number of particles (n) do

Randomly choose one candidate from C

eq.pool

Randomly generate the values of λ and r between [0, 1]

Construct F=a

sign(r − 0.5)[e

λt

− 1],

Construct GCP =

(

0.5 r

≥ GP

0 r

≤ GP

Construct G

= GCP(C

− λC),

Construct G = G

.F,

Update Concentrations C = C

+ (C

− C

).F +

λV

(1 − F)

end

Iteration = Iteration + 1

end

The generation rate

is one of the most important terms in the EO algorithm for

providing the exact solution, which is achieved by improving the exploitation phase. In

engineering applications, there are many models that can be used to express the generation

rate as a function of time

. For example, one multi-purpose model that describes generation

rates as a ﬁrst-order exponential decay process is deﬁned as

−→

G =

−→

−

−→

k (t−t

)

(7)

where

is the initial value and

indicates a decay constant. To have a more controlled

and systematic search pattern, and to limit the number of random variables, this paper

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