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A multi-objective particle swarm for constraint and unconstrained problems

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Abstract

Multi-objective particle swarm optimization algorithms (MOPS) are used successfully to solve real-life optimization problems. The multi-objective algorithms based on particle swarm optimization (PSO) have seen various adaptations to improve convergence to the true Pareto-optimal front and well-diverse non-dominated solution. In some cases, the values of the MOPS control parameters need to be fine-tuned while solving a specific multi-objective optimization problem. It is challenge to correctly fine-tune the value of the PSO control parameters when the true non-dominated solutions are not known as in case of a real-life optimization problem. To address this challenge, a multi-objective particle swarm optimization algorithm that uses constant PSO control parameters was developed. The new algorithm called NF-MOPSO is capable of solving different multi-objective optimization problems without the need of fine-tuning the value of the PSO control parameters. The NF-MOPSO enhances the convergence to the true Pareto-optimal front and improves the diversity of Pareto-optimal using the same fixed values for all the PSO control parameters. The NF-MOPSO uses constant values of the PSO control parameters such as acceleration coefficients \(c_{1}\) and \(c_{2}\), and inertia weight \(\omega\). A Gaussian mutation is applied to the position of particles to increase diversity while a penalty function is used as constraint mechanism. The algorithm has been tested on 45 well-known benchmark test functions using four performance metrics. The test results demonstrate the capability of the NF-MOPSO to solve different multi-objective optimization problems using the same value of the PSO control parameters. The capability of the NF-MOPSO was demonstrated in real-life optimization problem by solving a multi-objective optimization problem of a neutron radiography collimator. The results of collimator optimization showed that the optimizer was able to provide a set of Pareto optimal solutions from which the geometrical design parameters of a collimator could be retrieved for given application.

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Acknowledgements

The author wishes to thank the financial support of the South African Nuclear Energy Corporation and the National Research Fund.

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Authors and Affiliations

  1. Department of Industrial Engineering, University of Stellenbosch, Stellenbosch, South Africa

    Robert Nshimirimana

  2. Machine Intelligence Research Labs (MIR Labs), Scientific Network for Innovation and Research Excellence Auburn, Washington, 98071, USA

    Ajith Abraham

  3. Radiation Science Department, South African Nuclear Energy Corporation SOC Ltd, Pretoria, South Africa

    Robert Nshimirimana & Gawie Nothnagel

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Appendices

Appendix 1: NF-MOPSO algorithm

See Fig. 8.

Fig. 8
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NF-MOPSO algorithm

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Appendix 2: Performance results of two objectives test functions

See Figs. 9, 10, 11, 12, 13, 14.

Fig. 9
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Graphical performance results for two objectives test functions of NF-MOPSO vs true

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Fig. 10
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Graphical performance results for two objectives test functions of NF-MOPSO vs Theory

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Fig. 11
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Graphical performance results for two objectives test functions of NF-MOPSO vs Theory

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Fig. 12
figure 12

Graphical performance results for two objectives test functions of NF-MOPSO vs Theory

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Fig. 13
figure 13

Graphical performance results for two objectives test functions of NF-MOPSO vs Theory

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Fig. 14
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Graphical performance results for two objectives test functions of NF-MOPSO vs Theory

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Appendix 3: Performance results of three objectives test functions

See Figs. 15, 16.

Fig. 15
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Graphical performance results for three objectives test functions of NF-MOPSO vs Theory

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Fig. 16
figure 16

Graphical performance results for three objectives test functions of NF-MOPSO vs Theory

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Appendix 4: Flowchart of collimator optimization

See Fig. 17.

Fig. 17
figure 17

Flowchart of collimator optimization

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Nshimirimana, R., Abraham, A. & Nothnagel, G. A multi-objective particle swarm for constraint and unconstrained problems. Neural Comput & Applic 33, 11355–11385 (2021). https://doi.org/10.1007/s00521-020-05555-6

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