This study proposes a hybridization of two efficient algorithm’s Multi-objective Ant Lion Optimizer Algorithm (MOALO) which is a multi-objective enhanced version of the Ant Lion Optimizer Algorithm (ALO) and the Genetic Algorithm (GA). MOALO version has been employed to address those problems containing many objectives and an archive has been employed for retaining the non-dominated solutions. The uniqueness of the hybrid is that the operators like mutation and crossover of GA are employed in the archive to update the solutions and later those solutions go through the process of MOALO. A first-time hybrid of these algorithms is employed to solve multi-objective problems. The hybrid algorithm overcomes the limitation of ALO of getting caught in the local optimum and the requirement of more computational effort to converge GA. To evaluate the hybridized algorithm’s performance, a set of constrained, unconstrained test problems and engineering design problems were employed and compared with five well-known computational algorithms-MOALO, Multi-objective Crystal Structure Algorithm (MOCryStAl), Multi-objective Particle Swarm Optimization (MOPSO), Multi-objective Multiverse Optimization Algorithm (MOMVO), Multi-objective Salp Swarm Algorithm (MSSA). The outcomes of five performance metrics are statistically analyzed and the most efficient Pareto fronts comparison has been obtained. The proposed hybrid surpasses MOALO based on the results of hypervolume (HV), Spread, and Spacing. So primary objective of developing this hybrid approach has been achieved successfully. The proposed approach demonstrates superior performance on the test functions, showcasing robust convergence and comprehensive coverage that surpasses other existing algorithms.
In recent years, to solve difficult problems, computers have recently gained a lot of popularity across a variety of industries. The use of computers to solve issues and develop systems is the focus of the field of computer-aided design. In the prior, direct human engagement was needed for solving complex problems. More time and resources were also needed. Similarly, there has been a significant advancement in the field of optimization.
Gaining the most from the other options is what is meant by optimization. Based on the number of objectives taken, optimization problems are mainly classified as Single objective optimization problems (SOOPs) and Multi-objective optimization problems (MOOPs). SOOPs are the problems having one objective and MOOPs have more than one objective. In SOOPs, a single solution, however for MOOPs a set of trade-off solutions called Pareto-optimal solutions is obtained. Edgeworth and Pareto developed the optimality notion for MOOPs in their books in 1881 and 1906, respectively [
The two key components of Metaheuristic algorithms are exploration and exploitation. The perfect balance between them is the performance assessment of any algorithm in solving given MOOPs. Exploration refers to the search for the unexplored area and exploitation refers to the search for the immediate area to deliver an accurate search and convergence.
No Free Lunch theorem (NFL) for optimization encourages experts to develop the algorithms and enhance the ones that are already in use [
This paper proposes the hybrid of MOALO and GA for solving MOOPs. As best, this is the first comprehensive hybrid between MOALO and GA employed to solve MOOPs and design problems. According to the NFL, one single algorithm cannot have the expertise to solve all the existing problems. Real-world problems are more complex nowadays and also have many objectives to satisfy, so to solve those problems more and more algorithms are needed. For handling challenging real-world situations, metaheuristic and hybrid metaheuristic algorithms are preferable. Inspired by this, two metaheuristics have been chosen MOALO and GA, this hybrid will try to overcome the limitation of MOALO of efficiency and accuracy and GA’s computational effort to converge.
This work’s primary contributions can be specified as follows:
A hybridization strategy that combines MOALO and GA has been put forward as a means of tackling MOOPs and mitigating the limitations of each method.
Next, the use of performance metrics like generational distance (GD), inverted generational distance
Hybrid has been subjected to testing on constrained, unconstrained benchmark problems and design problems.
The remaining portion of the paper is presented as
A general MOOP [
Subject to:
where Z-Number of objectives, D-Inequality constraints,
Different algorithms have been employed to resolve MOOPs including the Multi-objective Bat algorithm [
Year | Algorithms |
---|---|
2014 | Particle Swallow Swarm Optimization (HPSSO) [ |
2015 | PSO and Estimation of Distribution Algorithm (EDA) [ |
2017 | Backtracking Search and Genetic Algorithm (HMOBSA) [ |
2020 | Gravitational Search Algorithm and BAT Algorithm (MOGSABAT) [ |
2020 | Spotted Hyena Optimizer and Emperor Penguin Optimizer (MOSHEPO) [ |
2021 | ALO Algorithm and Sine Cosine Algorithm (SCA) [ |
2022 | GWO and Support Vector Machine (SVM) [ |
2022 | ABC and DE (HABC-DE) [ |
2023 | Hybrid Multi-Objective Evolutionary Algorithm (MOEA) [ |
Some of the hybrid algorithms are as follows:
Seyedali Mirjalili established the ant lion algorithm in 2015. The ALO algorithm is inspired by antlion and ant interaction in the trap. To resolve MOOPs, MOALO [
The fundamental ALO processes to modify ants and antlions position and finally calculate the global optimum for a specific problem of optimization are:
1. Ants are the prime agents of search in ALO. So, in the initialization process, random values are assigned for the ants set.
2. At each iteration each ant’s fitness value is measured by the use of an objective function.
3. It is assumed that antlions are at one place and ants move randomly around antlions and over the search space.
4. Antlion populations are never determined. In the first iteration, antlions are expected to be where the ants are, but as the ants improve, they move to new positions in successive iterations.
5. Each ant is assigned an antlion and the position gets updated as the ant becomes fitter.
6. The best antlion obtained is defined as elite and affects the ant’s movement.
7. By the successive iterations, any antlion that outperforms the elite will get replaced by the elite.
8. The steps numbered 2 and 7 are executed iteratively multiple times until a certain condition is met, which serves as the termination criterion for the algorithm.
9. For the global optimum value the elite antlion’s position and fitness value are considered.
The changes made to MOALO set it apart from ALO since they used archives to store Pareto optimum solutions. To solve this issue of identifying Pareto optimal solutions with large diversity, “leader selection and archive maintenance” have been used. Concerning assessing the distribution of solutions within the archive, the niching technique was utilized. A further modification was the use of the roulette wheel selection and elitism, which was employed in selecting a non-dominated solution.
Genetic algorithm (GA) was first discovered by Holland and extensions by Goldberg in the 1960s influenced by Darwin’s evolutionary theory [
Basic operators involved in GA for finding the optimal solutions are defined as:
MOALG is developed in this paper which is a hybridised algorithm. The population of ants and antlions is generated randomly in the proposed hybrid like ALO [
This section involves the steps utilized in the proposed hybrid MOALG. The following steps are:
The population of ant and antlion is generated by
The ant’s random walk (RW) is entitled as:
where iteration is illustrated by t as RW steps
The stochastic function is illustrated as r(t)
r(t) is
where at the iteration
By modifying the RW of ants around antlions, ALO facilitates the entrapment of ants in antlion pits. The equation proposed for this is as follows:
To simulate the movement of ants towards antlions in a simulation using random walks, it is often necessary to decrease the boundaries of the random walks adaptively. This adaptive adjustment of the boundaries is done to simulate the ants sliding over to the antlion.
where
I is a ratio. t, T are the current and maximum number of iterations, respectively.
The parameter “w” in the
w | t |
---|---|
2 | >10% T |
3 | >50% T |
4 | >75% T |
5 | >90% T |
6 | >95% T |
The next step is to update the position after catching the ants. The equation below is expressed in this context:
where t is determined by the current iteration and i^{th} and j^{th} are ant and selected antlion’s position, respectively.
The final phase is elitism, where the antlion having the fittest value is saved during optimization at each iteration. The following process is shown:
The solutions with the fewer neighboring solutions are chosen for the antlions.
The solution with the densely packed neighboring is discarded when the archive is full to free up space for new solutions.
where c > 1 and is a constant. At i^{th} solution,
After finding the fitness of each ant and elite. In MOALG, the Archive has been updated by applying the ranking process. From the archive, two solutions are selected as parent 1 and parent 2 by the roulette wheel selection. The new generation (offspring) is generated by applying a one-point crossover operator with a certain probability
After that, elements of the new offspring mutate at a certain probability
A random walk around antlion includes steps
In this section, the MOALG algorithm’s efficiency for test problem having many objectives, and unconstrained and constrained problem have been assessed. These problems are employed to test the MOO handling of non-convex and non-linear situations. The algorithm was programmed in MATLAB 2021. To complete the work at that point, the computer’s following features are employed: 8 GB of RAM of Intel Core i9 with CPU (1.19 GHz).
When analyzing the Pareto-optimal solutions those retrieved by the MOO techniques are frequently evaluated using the following criteria [
1. Uniformity: The
2. Convergence: The solutions that come closest to the
3. Coverage: The exact solution should cover the
Veldhuizen (1998) addressed the GD [
where
It is common practice to compare meta-heuristics using IGD [
Spacing [
where in the OPF,
Deb in 2002 [
where
A smaller value of spread denotes a more diverse and even distribution of the non-dominated solutions.
Brest in 2006 [
K-non-dominated set of solutions and
For, each
The results of MOALO, MSSA, MOPSO, MOMVO, and MOCryStAl have been compared to MOALG. The Pareto optimal curve of MOALO and MOALG is shown in
Parameters | MOALG | MOALO | MOPSO | MSSA | MOMVO | MOCryStAl |
---|---|---|---|---|---|---|
Population size ( |
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Archive size ( |
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Global learning coefficient ( |
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Personal learning coefficient ( |
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Inertia weight (w) | ||||||
Beta | ||||||
Gamma |
This section examines the performance of existing algorithms on various benchmark functions constrained and unconstrained problems for performance metrics such as spread, spacing, HV, GD, and IGD. MOALG outperforms various algorithms in GD and IGD results. Spread, spacing, and HV results indicate convergence and diversity, MOALG surpasses the MOALO algorithm on various benchmark functions.
Functions | Algorithms | |||||
---|---|---|---|---|---|---|
MOALG | MOALO | MOPSO | MSSA | MOMVO | MOCrystal | |
5.3281E-06 | 1.7932E-02 | 1.2035E-02 | 7.9682E-03 | 2.5915E-04 | ||
1.9638E-05 | 5.0496E-03 | 1.9168E-03 | 1.2073E-02 | 4.0091E-04 | ||
9.2704E-04 | 1.4749E-01 | 1.1652E-02 | 5.3988E-03 | 1.6997E-04 | ||
4.7350E-04 | 6.5211E-02 | 2.1916E-03 | 2.9609E-04 | 4.7636E-04 | ||
3.6388E-05 | 2.0489E-04 | 1.6015E-02 | 1.0954E-02 | 2.0043E-04 | ||
1.9542E-04 | 1.1107E-04 | 2.0859E-03 | 1.5715E-02 | 1.6713E-04 | ||
8.00E-03 | 4.1299E+00 | 6.0099E-01 | 1.2363E+00 | 1.8557E+01 | ||
1.08E-02 | 3.9519E+00 | 3.1030E-01 | 7.0133E-01 | 1.1926E+01 | ||
6.2661E-04 | 2.5805E-02 | 2.1047E-02 | 1.0183E-02 | 1.0071E-02 | ||
4.20E-03 | 5.8008E-02 | 9.6106E-03 | 8.4995E-03 | 3.0359E-03 | ||
9.7941E-05 | 8.3098E-04 | 1.6878E-03 | 9.2588E-04 | 1.5219E-03 | ||
1.40E-03 | 6.3863E-04 | 5.8950E-04 | 2.4850E-04 | 5.9590E-04 |
Functions | Algorithms | |||||
---|---|---|---|---|---|---|
MOALG | MOALO | MOPSO | MSSA | MOMVO | MOCrystal | |
2.3977E-04 | 1.00E-03 | 8.0879E-04 | 4.5908E-03 | 1.5496E-03 | ||
3.4539E-04 | 1.3E-03 | 1.4764E-03 | 1.1201E-03 | 1.0733E-04 | ||
8.70E-03 | 5.2023E-02 | 5.2059E-03 | 1.9238E-03 | 1.6173E-03 | ||
0 | 1.10E-03 | 9.5007E-03 | 1.6220E-03 | 5.1549E-04 | ||
2.80E-03 | 2.6241E-04 | 5.1290E-03 | 1.6411E-03 | 1.3612E-03 | ||
2.50E-03 | 3.9884E-05 | 8.9560E-04 | 2.1512E-04 | 1.7230E-03 | ||
9.29E-02 | 7.0747E-01 | 1.8230E-01 | 2.4965E-01 | 8.8854E-01 | ||
1.14E-02 | 3.1784E-01 | 9.5457E-01 | 1.3922E-01 | 1.0955E-01 | ||
2.30E-03 | 8.2281E-03 | 2.0670E-03 | 5.0062E-04 | 3.5820E-04 | ||
6.6194E-04 | 1.10E-03 | 2.4862E-02 | 8.0527E-04 | 1.9857E-04 | ||
9.60E-03 | 5.1838E-04 | 2.1706E-03 | 1.8516E-03 | 1.1648E-03 | ||
4.1632E-04 | 9.60E-03 | 7.3799E-04 | 3.6526E-04 | 2.6604E-04 |
In
Algorithms | ZDT1 | ZDT2 | ZDT3 | ZDT4 | ZDT6 | CONSTR |
---|---|---|---|---|---|---|
1.17E-02 | 1.17E-02 | 1.08E-02 | 1.02E-02 | 1.53E-02 | 1.22E-02 |
Algorithms | ZDT1 | ZDT2 | ZDT3 | ZDT4 | ZDT6 | CONSTR |
---|---|---|---|---|---|---|
1.1853E-04 | 7.6394E-04 | 8.30E-03 | 4.20E-03 | 1.7E-03 | ||
1.7138E-04 |
Algorithms | ZDT1 | ZDT2 | ZDT3 | ZDT4 | ZDT6 | CONSTR |
---|---|---|---|---|---|---|
0 | 3.30E-03 | |||||
6.10E-03 | 7.60E-03 | 0 | 9.0E-03 |
Two design problems have been solved. Four-bar truss design problem [
Algorithms | GD | IGD | Spread | Spacing | HV |
---|---|---|---|---|---|
(Mean & SD) | (Mean & SD) | (Mean) | (Mean) | (Score) | |
1.264E-01 | 1.062E-01 | 3.70E-01 | 1.18E+00 | 3.70E-02 | |
3.27E-02 | 1.52E-02 | ||||
1.4095E+01 | 2.0010E-02 | 1.4876E+00 | 5.3605E+00 | 1.16E-04 | |
5.1580E-01 | 3.9632E-05 | ||||
7.7966E+00 | 2.1434E-02 | 1.2019E+00 | 6.1160E+00 | 8.90E-04 | |
3.5486E+00 | 3.8292E-04 | ||||
1.1017E+01 | 2.1020E-02 | 1.4014E+00 | 4.8245E+00 | 6.17E-04 | |
4.6552E+00 | 4.6879E-04 | ||||
6.8970E+00 | 2.0005E-02 | 1.5436E+00 | 1.1012E+00 | 5.69E-05 | |
1.2099E+00 | 3.5863E-05 |
Algorithms | GD | IGD | Spread | Spacing | HV |
---|---|---|---|---|---|
(Mean & SD) | (Mean & SD) | (Mean) | (Mean) | (Score) | |
5.62E-01 | |||||
6.65E-03 | 1.52E-03 | 1.978E-01 | 4.26E-02 | 1.49E-03 | |
7.42E-03 | 4.65E-03 | ||||
1.1946E-02 | 5.9705E-04 | 1.0072E+00 | 2.3432E-01 | 1.16E-04 | |
1.9956E-03 | 4.6341E-05 | ||||
6.3467E-03 | 4.8242E-03 | 7.9085E-01 | 1.90E-04 | ||
3.0983E-03 | 3.5852E-03 | ||||
1.5031E-02 | 2.1075E-03 | 1.0552E+00 | 2.0967E-01 | 2.17E-04 | |
4.7477E-03 | 4.2017E-04 | ||||
6.6659E-02 | 1.3180E-03 | 1.0709E+00 | 3.6825E-01 | 1.69E-03 | |
1.1624E-01 | 3.4394E-04 |
Symbol | Description |
---|---|
Upper bound | |
Lower bound | |
p | Maximum number of iterations |
Cumulative sum | |
RW | Random walk |
Maximum of RW in i^{th} variable | |
Minimum of RW in i^{th} variable | |
Maximum of all variable at t^{th} iteration | |
Minimum of all variable at t^{th} iteration | |
RW around the antlion | |
RW around the elite | |
Position of i^{th} at t^{th} iteration | |
Pareto optimal front |
This paper proposed a hybrid of MOALO and GA named MOALG. Selection mechanism and crossover, mutation operator have been used for bettering of non-dominated solutions in the archive. The primary objective behind the development of MOALG is to overcome the limitations of ALO of getting caught in a local optimum and more computational effort to converge GA. The algorithm tests various problems using performance metrics such as GD, IGD, Spread, Spacing, and HV. For the comparison, the other competitive algorithms were utilized as MOALO, MOPSO, MSSA, MOMVO, and MOCryStAl. MOALG has been observed to benefit from strong convergence and coverage. MOALG has performed better than MOALO which can be seen through the numerical and statistical results. It can be viewed that MOALG can also find the Pareto optimal front of any shape. Another conclusion is that as per the NFL theorem, MOALG can perform better for other problems as well.
For future work, MOALG can be tested for various engineering design problems and many other real-life problems of multi-objectives.
The authors wish to acknowledge the resources of Chandigarh University, Mohali, India, Soonchunhyang University, Korea, Michigan State University, USA, and Mansoura University, Egypt for carrying out the research work.
This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIT) (No. RS-2023-00218176), and the Soonchunhyang University Research Fund.
The authors confirm contribution to the paper as follows: study conception and design: R. Sharma, A. Pal, N. Mittal; data collection: S. Van, L. Kumar; analysis and interpretation of results: R. Sharma, Y. Nam; draft manuscript preparation: L. Kumar, R. Sharma, A. Pal, M. Abouhawwash. All authors reviewed the results and approved the final version of the manuscript.
Data is available from the authors upon reasonable request from the authors.
The authors declare that they have no conflicts of interest to report regarding the present study.
A convex Pareto-front is obtained here equation containing two constraints:
This is a well-known field in structural optimization were
Minimize:
where