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Multi-objective ant lion optimizer: a multi-objective optimization algorithm for solving engineering problems

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Abstract

This paper proposes a multi-objective version of the recently proposed Ant Lion Optimizer (ALO) called Multi-Objective Ant Lion Optimizer (MOALO). A repository is first employed to store non-dominated Pareto optimal solutions obtained so far. Solutions are then chosen from this repository using a roulette wheel mechanism based on the coverage of solutions as antlions to guide ants towards promising regions of multi-objective search spaces. To prove the effectiveness of the algorithm proposed, a set of standard unconstrained and constrained test functions is employed. Also, the algorithm is applied to a variety of multi-objective engineering design problems: cantilever beam design, brushless dc wheel motor design, disk brake design, 4-bar truss design, safety isolating transformer design, speed reduced design, and welded beam deign. The results are verified by comparing MOALO against NSGA-II and MOPSO. The results of the proposed algorithm on the test functions show that this algorithm benefits from high convergence and coverage. The results of the algorithm on the engineering design problems demonstrate its applicability is solving challenging real-world problems as well.

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References

  1. Kelley CT (1999) Detection and remediation of stagnation in the Nelder–Mead algorithm using a sufficient decrease condition. SIAM J Optim 10:43–55

    Article  MathSciNet  MATH  Google Scholar 

  2. Vogl TP, Mangis J, Rigler A, Zink W, Alkon D (1988) Accelerating the convergence of the back-propagation method. Biol Cybern 59:257–263

    Article  Google Scholar 

  3. Marler RT, Arora JS (2004) Survey of multi-objective optimization methods for engineering. Struct Multidiscip Optim 26:369–395

    Article  MathSciNet  MATH  Google Scholar 

  4. Coello CAC (2002) Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art. Comput Methods Appl Mech Eng 191:1245–1287

    Article  MathSciNet  MATH  Google Scholar 

  5. Beyer H.-G., Sendhoff B (2007) Robust optimization–a comprehensive survey. Comput Methods Appl Mech Eng 196:3190–3218

    Article  MathSciNet  MATH  Google Scholar 

  6. Knowles JD, Watson RA, Corne DW (2001) Reducing local optima in single-objective problems by multi-objectivization. In: Evolutionary multi-criterion optimization, pp 269–283

  7. Deb K, Goldberg DE (1993) Analyzing deception in trap functions. Found Genet Algoritm 2:93–108

    Article  Google Scholar 

  8. Deb K (2001) Multi-objective optimization using evolutionary algorithms, vol 16. Wiley

  9. Coello CAC, Lamont GB, Van Veldhuisen DA (2007) Evolutionary algorithms for solving multi-objective problems. Springer

  10. Branke J, Deb K, Dierolf H, Osswald M (2004) Finding knees in multi-objective optimization. In: Parallel problem solving from nature-PPSN VIII, pp 722–731

  11. Das I, Dennis JE (1998) Normal-boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J Optim 8:631–657

    Article  MathSciNet  MATH  Google Scholar 

  12. Kim IY, De Weck O (2005) Adaptive weighted-sum method for bi-objective optimization: Pareto front generation. Struct Multidiscip Optim 29:149–158

    Article  Google Scholar 

  13. Messac A, Mattson CA (2002) Generating well-distributed sets of Pareto points for engineering design using physical programming. Optim Eng 3:431–450

    Article  MATH  Google Scholar 

  14. Deb K, Agrawal S, Pratap A, Meyarivan T (2000) A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: NSGA-II. In: Parallel problem solving from nature PPSN VI, pp 849–858

  15. Deb K, Goel T (2001) Controlled elitist non-dominated sorting genetic algorithms for better convergence. In: Evolutionary multi-criterion optimization, pp 67–81

  16. Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6:182–197

    Article  Google Scholar 

  17. Coello CAC, Lechuga MS (2002) MOPSO: a proposal for multiple objective particle swarm optimization. In: Proceedings of the 2002 congress on evolutionary computation, 2002. CEC’02, pp 1051–1056

  18. Coello CAC, Pulido GT, Lechuga MS (2004) Handling multiple objectives with particle swarm optimization. IEEE Trans Evol Comput 8:256–279

    Article  Google Scholar 

  19. Akbari R, Hedayatzadeh R, Ziarati K, Hassanizadeh B (2012) A multi-objective artificial bee colony algorithm. Swarm Evol Comput 2:39–52

    Article  Google Scholar 

  20. Yang X-S (2011) Bat algorithm for multi-objective optimisation. Int J Bio-Inspired Comput 3:267–274

    Article  Google Scholar 

  21. Mirjalili S, Saremi S, Mirjalili SM, Coelho L. d. S. (2016) Multi-objective grey wolf optimizer: a novel algorithm for multi-criterion optimization. Expert Syst Appl 47:106–119

    Article  Google Scholar 

  22. Wolpert DH, Macready WG (1997) No free lunch theorems for optimization. IEEE Trans Evol Comput 1:67–82

    Article  Google Scholar 

  23. Ngatchou P, Zarei A, El-Sharkawi M (2005) Pareto multi objective optimization. In: Proceedings of the 13th international conference on intelligent systems application to power systems, 2005, pp 84–91

  24. Pareto V (1964) Cours d’economie politique: Librairie Droz

  25. Edgeworth FY (1881) Mathematical physics. P. Keagan, London

    Google Scholar 

  26. Zitzler E (1999) Evolutionary algorithms for multiobjective optimization: methods and applications, vol 63. Citeseer

  27. Zitzler E, Thiele L (1999) Multiobjective evolutionary algorithms: a comparative case study and the strength pareto approach. IEEE Trans Evol Comput 3:257–271

    Article  Google Scholar 

  28. Srinivas N, Deb K (1994) Muiltiobjective optimization using nondominated sorting in genetic algorithms. Evol Comput 2:221–248

    Article  Google Scholar 

  29. Zhang Q, Li H (2007) MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evol Comput 11:712–731

    Article  Google Scholar 

  30. Knowles JD, Corne DW (2000) Approximating the nondominated front using the Pareto archived evolution strategy. Evol Comput 8:149–172

    Article  Google Scholar 

  31. Abbass HA, Sarker R, Newton C (2001) PDE: a Pareto-frontier differential evolution approach for multi-objective optimization problems. In: Proceedings of the 2001 congress on evolutionary computation, 2001, pp 971–978

  32. Branke J, Kaußler T, Schmeck H (2001) Guidance in evolutionary multi-objective optimization. Adv Eng Softw 32:499– 507

    Article  MATH  Google Scholar 

  33. Eberhart RC, Kennedy J (1995) A new optimizer using particle swarm theory. In: Proceedings of the sixth international symposium on micro machine and human science, pp 39– 43

  34. Kennedy J (2011) Particle swarm optimization. In: Encyclopedia of machine learning. Springer, pp 760–766

  35. Horn J, Nafpliotis N, Goldberg DE (1994) A niched Pareto genetic algorithm for multiobjective optimization. In: Proceedings of the 1st IEEE conference on evolutionary computation, 1994. IEEE world congress on computational intelligence, pp 82–87

  36. Mahfoud SW (1995) Niching methods for genetic algorithms. Urbana 51:62–94

    Google Scholar 

  37. Horn JR, Nafpliotis N, Goldberg DE (1993) Multiobjective optimization using the niched pareto genetic algorithm. IlliGAL report, pp 61801–2296

  38. Mirjalili S (2015) The ant lion optimizer. Adv Eng Softw 83:80–98

    Article  Google Scholar 

  39. Van Veldhuizen DA, Lamont GB (1998) Multiobjective evolutionary algorithm research: a history and analysis. Citeseer

  40. Sierra MR, Coello CAC (2005) Improving PSO-based multi-objective optimization using crowding, mutation and ∈-dominance. In: Evolutionary multi-criterion optimization, pp 505–519

  41. Schott JR (1995) Fault tolerant design using single and multicriteria genetic algorithm optimization, DTIC Document

  42. Zitzler E, Deb K, Thiele L (2000) Comparison of multiobjective evolutionary algorithms: Empirical results. Evolutionary computation 8(2):173–195

    Article  Google Scholar 

  43. Mirjalili S (2016) Dragonfly algorithm: a new meta-heuristic optimization technique for solving single-objective, discrete, and multi-objective problems. Neural Comput & Applic 27:1053–1073

    Article  Google Scholar 

  44. Coello CAC (2000) Use of a self-adaptive penalty approach for engineering optimization problems. Comput Ind 41:113– 127

    Article  Google Scholar 

  45. Sadollah A, Eskandar H, Kim JH (2015) Water cycle algorithm for solving constrained multi-objective optimization problems. Appl Soft Comput 27:279–298

    Article  Google Scholar 

  46. Srinivasan N, Deb K (1994) Multi-objective function optimisation using non-dominated sorting genetic algorithm. Evol Comput 2:221–248

    Article  Google Scholar 

  47. Binh TT, Korn U (1997) MOBES: A multiobjective evolution strategy for constrained optimization problems. In: The 3rd international conference on genetic algorithms (Mendel 97), p 7

  48. Osyczka A, Kundu S (1995) A new method to solve generalized multicriteria optimization problems using the simple genetic algorithm. Struct Optim 10:94–99

    Article  Google Scholar 

  49. Coello CC, Pulido GT (2005) Multiobjective structural optimization using a microgenetic algorithm. Struct Multidiscip Optim 30:388–403

    Article  Google Scholar 

  50. Kurpati A, Azarm S, Wu J (2002) Constraint handling improvements for multiobjective genetic algorithms. Struct Multidiscip Optim 23:204–213

    Article  Google Scholar 

  51. Ray T, Liew KM (2002) A swarm metaphor for multiobjective design optimization. Eng Optim 34:141–153

    Article  Google Scholar 

  52. Moussouni F, Brisset S, Brochet P (2007) Some results on the design of brushless DC wheel motor using SQP and GA. Int J Appl Electromagn Mech 26:233–241

    Google Scholar 

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Author information

Authors and Affiliations

  1. School of Information and Communication Technology, Griffith University, Nathan Campus, Brisbane, QLD, 4111, Australia

    Seyedali Mirjalili & Shahrzad Saremi

  2. Griffith College, Mt Gravatt, Brisbane, QLD, 4122, Australia

    Seyedali Mirjalili & Shahrzad Saremi

  3. Lukhdhirji Engineering College, Morbi-Rajkot, Gujarat, India

    Pradeep Jangir

Authors
  1. Seyedali Mirjalili
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  2. Pradeep Jangir
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  3. Shahrzad Saremi
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Corresponding author

Correspondence to Seyedali Mirjalili.

Appendices

Appendix A: Unconstrained multi-objective test problems utilised in this work

ZDT1:

$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{1}(x)=x_{1} \end{array} $$
(A.1)
$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{2}(x)=g(x)\times h(f_{1}(x),g(x)) \end{array} $$
(A.2)
$$\begin{array}{@{}rcl@{}} \text{Where:}{\kern12pt}&~&G(x)=1+\frac{9}{N-1}\sum\limits_{i=2}^{N}x_{i} \end{array} $$
(A.3)
$$\begin{array}{@{}rcl@{}} &~&h(f_{1}(x),g(x))=1-\sqrt{\frac{f_{1}(x)}{g(x)}}\\ &~&0\leq x_{i}\leq1,1\leq i\leq30 \end{array} $$
(A.4)

ZDT2:

$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{1}(x)=x_{1} \end{array} $$
(A.5)
$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{2}(x)=g(x)\times h(f_{1}(x),g(x)) \end{array} $$
(A.6)
$$\begin{array}{@{}rcl@{}} \text{Where:}{\kern12pt}&~&G(x)=1+\frac{9}{N-1}\sum\limits_{i=2}^{N}x_{i} \end{array} $$
(A.7)
$$\begin{array}{@{}rcl@{}} &~&h(f_{1}(x),g(x))=1-\left( {\frac{f_{1}(x)}{g(x)}}\right)^{2}\\ &~&0\leq x_{i}\leq1,1\leq i\leq30 \end{array} $$
(A.8)

ZDT3:

$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{1}(x)=x_{1} \end{array} $$
(A.9)
$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{2}(x)=g(x)\times h(f_{1}(x),g(x)) \end{array} $$
(A.10)
$$\begin{array}{@{}rcl@{}} \text{Where:}{\kern12pt}&~&G(x)=1+\frac{9}{29}\sum\limits_{i=2}^{N}x_{i} \end{array} $$
(A.11)
$$\begin{array}{@{}rcl@{}} &~&h(f_{1}(x),g(x))=1-\sqrt{\frac{f_{1}(x)}{g(x)}} \end{array} $$
(A.12)
$$\begin{array}{@{}rcl@{}}&&\qquad\qquad-\left( {\frac{f_{1}(x)}{g(x)}}\right)\sin(10\pi f_{1}(x))\\ &~&0\leq x_{i}\leq1,1\leq i\leq30 \end{array} $$

ZDT1 with linear PF:

$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{1}(x)=x_{1} \end{array} $$
(A.13)
$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{2}(x)=g(x)\times h(f_{1}(x),g(x)) \end{array} $$
(A.14)
$$\begin{array}{@{}rcl@{}} \text{Where:}{\kern12pt}&~&G(x)=1+\frac{9}{N-1}\sum\limits_{i=2}^{N}x_{i} \end{array} $$
(A.15)
$$\begin{array}{@{}rcl@{}} &~&h(f_{1}(x),g(x))=1-{\frac{f_{1}(x)}{g(x)}} \end{array} $$
(A.16)
$$\begin{array}{@{}rcl@{}} &~&0\leq x_{i}\leq1,1\leq i\leq30 \end{array} $$

ZDT2 with three objectives:

$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{1}(x)=x_{1} \end{array} $$
(A.17)
$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{2}(x)=x_{2} \end{array} $$
(A.18)
$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{3}(x)=g(x)\times h(f_{1}(x),g(x)) \end{array} $$
(A.19)
$$\begin{array}{@{}rcl@{}} &~&\qquad\qquad\times h(f_{2}(x),g(x))\\ \text{Where:}{\kern12pt}&~&G(x)=1+\frac{9}{N-1}\sum\limits_{i=2}^{N}x_{i} \end{array} $$
(A.20)
$$\begin{array}{@{}rcl@{}} &~&h(f_{1}(x),g(x))=1-\left( {\frac{f_{1}(x)}{g(x)}}\right)^{2} \end{array} $$
(A.21)
$$\begin{array}{@{}rcl@{}} &~&0\leq x_{i}\leq1,1\leq i\leq30 \end{array} $$

Appendix B: Constrained multi-objective test problems utilised in this work

CONSTR:

This problem has a convex Pareto front, and there are two constraints and two design variables.

$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{1}(x)=x_{1} \end{array} $$
(B.1)
$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{2}(x)=(1+x_{2})/(x_{1}) \end{array} $$
(B.2)
$$\begin{array}{@{}rcl@{}} \text{Where:}{\kern12pt}&~&g_{1}(x)\,=\,6-(x_{2}+9x_{1}),g_{2}(x)\,=\,1+x_{2}-9x_{1}\\ &~&0.1\leq x_{1}\leq 1,0\leq x_{2}\leq 5 \end{array} $$

TNK:

The second problem has a discontinuous Pareto optima front, and there are two constraints and two design variables.

$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{1}(x)=x_{1} \end{array} $$
(B.3)
$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{2}(x)=x_{2}\\ \text{Where:}{\kern12pt}&~&g_{1}(x)\,=\,-x_{1}^{\;2}\,-\,x_{2}^{\;2}\,+\,1\,+\,0.1 Cos\!\left( \!16arctan\left( \frac{x_{1}}{x_{2}}\right)\!\right)\\ &~&g_{2}(x)=0.5-(x_{1}-0.5)^{2}-(x_{2}-0.5)^{2}\\ &~&0.1\leq x_{1}\leq \pi,0\leq x_{2}\leq \pi \end{array} $$
(B.4)

SRN:

The third problem has a continuous Pareto optimal front proposed by Srinivas and Deb [46].

$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{1}(x)=2+(x_{1}-2)^{2}+(x_{2}-1)^{2} \end{array} $$
(B.5)
$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{2}(x)=9x_{1}-(x_{2}-1)^{2}\\ \text{Where:}{\kern12pt}&~&g_{1}(x)=x_{1}^{\;2}+x_{2}^{\;2}-255\\ &~&{}g_{2}(x)=x_{1}-3x_{2}+10\\ &~&{}-20\leq x_{1}\leq 20,-20\leq x_{2}\leq 20 \end{array} $$
(B.6)

BNH:

This problem was first proposed by Binh and Korn [47]:

$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{1}(x)=4x_{1}^{\;2}+4x_{2}^{\;2} \end{array} $$
(B.7)
$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{2}(x)=(x_{1}-5)^{2}+(x_{2}-5)^{2}\\ \text{Where:}{\kern12pt}&~&g_{1}(x)=(x_{1}-5)^{2}+x_{2}^{\;2}-25\\ &~&{}g_{2}(x)=7.7-(x_{1}-8)^{2}-(x_{2}+3)^{2}\\ &~&{}0\leq x_{1}\leq 5,0\leq x_{2}\leq 3 \end{array} $$
(B.8)

OSY:

The OSY test problem has five separated regions proposed by Osyczka and Kundu [48]. Also, there are six constraints and six design variables.

$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{1}(x)\,=\,x_{1}^{\;2}\,+\,x_{2}^{\;2}+x_{3}^{\;2}+x_{4}^{\;2}+x_{5}^{\;2}+x_{6}^{\;2} \end{array} $$
(B.9)
$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{2}(x)\,=\,[25(x_{1}\,-\,2)^{2}+(x_{2}\,-\,1)^{2}\,+\,(x_{3}-1)\\&~&\qquad\;\;\quad+(x_{4}-4)^{2}+(x_{5}-1)^{2}]\\ \text{Where:}{\kern12pt}&~&g_{1}(x)=2-x_{1}-x_{2}\\ &~&{}g_{2}(x)=-6+x_{1}+x_{2}\\ &~&{}g_{3}(x)=-2-x_{1}+x_{2}\\ &~&{}g_{4}(x)=-2+x_{1}-3x_{2}\\ &~&{}g_{5}(x)=-4+x_{4}+(x_{3}-3)^{2}\\ &~&{}g_{6}(x)=4-x_{6}-(x_{5}-3)^{2}\\ &~&{}0\leq x_{1}\leq 10,0\leq x_{2}\leq 10,1\leq x_{3}\leq5,0\leq x_{4}\\&~&{}\leq 6,1\leq x_{5}\leq5,0\leq x_{6}\leq 10 \end{array} $$
(B.10)

Appendix C: Constrained multi-objective engineering problems used in this work

1.1 Four-bar truss design problem

The 4-bar truss design problem is a well-known problem in the structural optimization field [49], in which structural volume (f 1) and displacement (f 2) of a 4-bar truss should be minimized. As can be seen in the following equations, there are four design variables (x 1- x 4) related to cross sectional area of members 1, 2, 3, and 4.

$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{1}(x)=200*(2*x(1)+sqrt(2*x(2))\\&~&{\kern32pt}+sqrt(x(3))+x(4)) \end{array} $$
(C.1)
$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{2}(x)=0.01*(\left( \frac{2}{x(1)}\right)+\left( \frac{2*sqrt(2)}{x(2)}\right)....\\ &~&{}-((2*sqrt(2))/x(3))+(2/x(1))) \end{array} $$
(C.2)
$$\begin{array}{@{}rcl@{}} 1\!\leq\! x_{1}\leq 3,1.4142\leq x_{2}\!\leq\! 3,1.4142\leq x_{3}\leq 3,1\leq x_{4}\leq 3 \end{array} $$

1.2 Speed reducer design problem

The speed reducer design problem is a well-known problem in the area of mechanical engineering [49, 50], in which the weight (f 1) and stress (f 2) of a speed reducer should be minimized. There are seven design variables: gear face width (x 1), teeth module (x 2), number of teeth of pinion (x 3 integer variable), distance between bearings 1 (x 4), distance between bearings 2 (x 5), diameter of shaft 1 (x 6), and diameter of shaft 2 (x 7) as well as eleven constraints.

$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{1}(x)=0.7854\!*\!x(1)\!*\!x(2)^{2}\!*\!(3.3333*x(3)^{2}\\ &&+14.9334*x(3))...\\ &~&{}-43.0934)-1.508*x(1)*(x(6)^{2}+x(7)^{2}\\ &~&{}+7.4777*(x(6)^{3}+x(7)^{3})...\\ &~&{}+0.7854*(x(4)*x(6)^{2}+x(5)*x(7)^{2}) \end{array} $$
(C.3)
$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{2}(x)=((sqrt(((745*x(4))/x(2)*x(3)))^{2}\\ &&+19.9e6))/(0.1*x(6)^{3})) \end{array} $$
(C.4)
$$\begin{array}{@{}rcl@{}} \text{Where:}{\kern12pt}&~&g_{1}(x)=27/(x(1)*x(2)^{2}*x(3))-1\\ &~&{}g_{2}(x)=397.5/(x(1)*x(2)^{2}*x(3)^{2})-1\\ &~&{}g_{3}(x)=(1.93*(x(4)^{3})/(x(2)*x(3)*x(6)^{4})-1\\ &~&{}g_{4}(x)=(1.93*(x(5)^{3})/(x(2)*x(3)*x(7)^{4})-1\\ &~&{}g_{5}(x)=((sqrt(((745 * x(4))/(x(2)*x(3)))^{2} \\ &~&{}+ 16.9e6))/(110 * x(6)^{3})) - 1\\ &~&{}g_{6}(x)=((sqrt(((745 * x(5))/(x(2)*x(3)))^{2} \\&~&{}+ 157.5e6))/(85 * x(7)^{3})) - 1\\ &~&{}g_{7}(x) = ((x(2) * x(3))/40) - 1\\ &~&{}g_{8}(x) = (5 * x(2)/x(1)) - 1\\ &~&{}g_{9}(x) = (x(1)/12* x(2))- 1\\ &~&{}g_{10}(x) = ((1.5 * x(6)+1.9)/x(4)) - 1\\ &~&{}g_{11}(x) = ((1.1 * x(7)+1.9)/x(5)) - 1\\ &~&{}2.6\leq x_{1}\leq3.6,0.7\leq x_{2}\leq0.8,17\leq x_{3}\leq 28,7.3\leq x_{4}\\&~&{}\leq 8.3,7.3\leq x_{5}\leq 8.3, 2.9\leq x_{6}\leq3.9\\ &~&{}5\leq x_{7}\leq5.5 \end{array} $$

1.3 Disk brake design problem

The disk brake design problem has mixed constraints and was proposed by Ray and Liew [51]. The objectives to be minimized are: stopping time (f 1) and mass of a brake (f 2) of a disk brake. As can be seen in following equations, there are four design variables: the inner radius of the disk (x 1), the outer radius of the disk (x 2), the engaging force (x 3), and the number of friction surfaces (x 4) as well as five constraints.

$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{1}(x)=4.9*(10^{(-5)})*(x(2)^{2}\,-\,x(1)^{2})*(x(4)\,-\,1)\\ \end{array} $$
(C.5)
$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{2}(x)=(9.82*(10^{(6)}))*(x(2)^{2}\\ &&\qquad{\kern12pt}-x(1)^{2}))/((x(2)^{3}-x(1)^{3})*x(4)*x(3))\\ \end{array} $$
(C.6)
$$\begin{array}{@{}rcl@{}} \text{Where:}{\kern12pt}&~&g_{1}(x)=20+x(1)-x(2)\\ &~&{}g_{2}(x)=2.5*(x(4)+1)-30\\ &~&{}g_{3}(x)=(x(3))/(3.14*(x(2)^{2}-x(1)^{2})^{2})-0.4\\ &~&{}g_{4}(x)=(2.22*10^{(-3)}*x(3)*(x(2)^{3}\\ &&{}-x(1)^{3}))/((x(2)^{2}-x(1)^{2})^{2})-1\\ &~&{}g_{5}(x)=900-(2.66*10^{(-2)}*x(3)*x(4)*(x(2)^{3}\\ &&{}-x(1)^{3}))/((x(2)^{2}-x(1)^{2}))\\ &~&{}55\!\leq\! x_{1}\!\leq\! 80,75\!\leq\! x_{2}\!\leq\! 110,1000\!\leq\! x_{3}\!\leq\! 3000,2\!\leq\!x_{4}\leq 20 \end{array} $$

1.4 Welded beam design problem

The welded beam design problem has four constraints first proposed by Ray and Liew [51]. The fabrication cost (f 1) and deflection of the beam (f 2) of a welded beam should be minimized in this problem. There are four design variables: the thickness of the weld (x 1), the length of the clamped bar (x 2), the height of the bar (x 3) and the thickness of the bar (x 4).

$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{1}(x)=1.10471*x(1)^{2}*x(2)\\&&\qquad\quad{\kern2pt}+0.04811*x(3)*x(4)*(14.0\,+\,x(2))\\ \end{array} $$
(C.7)
$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{2}(x)=65856000/(30*10^{6}*x(4)*x(3)^{3})\\ \text{Where:}{\kern12pt}&~&g_{1}(x)=\tau-13600\\ &~&{}g_{2}(x)=\sigma-30000\\ &~&{}g_{3}(x)=x(1)-x(4)\\ &~&{}g_{4}=6000-P\\ &~&{}0.125\leq x_{1}\leq 5,0.1\leq x_{2}\leq 10,0.1\leq x_{3}\leq10,0.125\leq x_{4}\leq 5 \end{array} $$
(C.8)

Where

$$\begin{array}{@{}rcl@{}} q&=&6000*\left( 14+\frac{x(2)}{2}\right); D=sqrt\left( \frac{x(2)^{2}}{4}\,+\,\frac{(x(1)+x(3))^{2}}{4}\right)\\ J&=&2*\left( x(1)*x(2)*sqrt(2)*\left( \frac{x(2)^{2}}{12}+\frac{(x(1)+x(3))^{2}}{4}\right)\right)\\ \alpha&=&\frac{6000}{sqrt(2)*x(1)*x(2)}\\ \beta&=&Q*\frac{D}{J} \end{array} $$
$$\begin{array}{@{}rcl@{}} &&{}\tau=sqrt\left( \alpha^{2}+2*\alpha*\beta*\frac{x(2)}{2*D}+\beta^{2}\right)\\ &&{}\sigma=\frac{504000}{x(4)*x(3)^{2}}\\ &&{}tmpf=4.013*\frac{30*10^{6}}{196}\\ &&{}P=tmpf*sqrt\left( x(3)^{2}*\frac{x(4)^{6}}{36}\right)*\left( 1-x(3)*\frac{sqrt\left( \frac{30}{48}\right)}{28}\right) \end{array} $$

1.5 Cantilever beam design problem

The cantilever beam design problem is another well-known problem in the field of concrete engineering [8], in which weight (f 1) and end deflection (f 2) of a cantilever beam should be minimized. There are two design variables: diameter (x 1) and length (x 2).

$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{1}(x)=0.25*\rho*\pi*x(2)*x(1)^{2} \end{array} $$
(C.9)
$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{2}(x)=(64*P*x(2)^{3})/(3*E*\pi*x(1)^{4})\\\\ \text{Where:}{\kern12pt}&~&g_{1}(x)=-Sy+(32*P*x(2))/(\pi*x(1)^{3})\\ &~&{}g_{2}(x)=-\delta_{max}+(64*P*x(2)^{3})/(3*E*\pi*x(1)^{4})\\ &~&{}0.01\leq x_{1}\leq 0.05, 0.20\leq x_{2}\leq 1 \end{array} $$
(C.10)

Where

$$P\,=\,1,E\,=\,207000000,Sy\,=\,300000,\delta_{max}\,=\,0.005;\rho\,=\,7800 $$

1.6 Brushless DC wheel motor with two objectives

Brushless DC wheel motor design problem is a constrained multi-objective problem in the area of electrical engineering [52]. The objectives are in conflict, and there are five design variables: stator diameter (D s), magnetic induction in the air gap (B e), current density in the conductors (δ), magnetic induction in the teeth (B d) and magnetic induction in the stator back iron (B c s).

$$\begin{array}{@{}rcl@{}} \text{Maximize:}&~&f_{1}(x)=\text{Max}\;\eta \end{array} $$
(C.11)
$$\begin{array}{@{}rcl@{}} \text{Minimize:}&~&f_{2}(x)=Min\;\;M_{tot}\\ &~&\qquad\;\; D_{ext}\leq340mm\\ \text{Where:}{\kern12pt}&~&G(x)=D_{\text{int}}\geq76mm, I_{\max}\geq125A\\ &~&\qquad\;\;T_{a}\leq120^{\circ},discr\geq0\\ &~&150mm\leq D_{s}\leq 330mm,0.5T\leq B_{e}\!\leq\!0.76T\\ &~&2A/mm^{2}\leq\sigma\!\leq\! 5A/mm^{2},0.9T\leq B_{d}\!\leq\!1.8T\\ &~&0.6T\leq B_{cs}\leq1.6T \end{array} $$
(C.12)

1.7 Safety isolating transformer design with two objectives

$$\begin{array}{@{}rcl@{}} \text{Maximize:}&~&f_{1}(x)=\text{Max}\;\;\eta \end{array} $$
(C.13)
$$\begin{array}{@{}rcl@{}} \text{Maximize:}&~&f_{2}(x)=Min\;\;M_{tot}\\ &~&T_{cond}\leq 120^{\circ}C, T_{iron}\leq 100^{\circ}C\\ \text{Where:}{\kern12pt}&~&G(x)=\frac{\Delta V_{2}}{V_{20}}\leq0.1,\frac{I_{10}}{I_{1}}\leq0.1\\ &~&f_{2}\leq1,f_{1}\leq1\\ &~&residue<10^{-6}\\ &~&3mm\leq a\leq30mm,14mm\leq b\leq95mm\\ &~&6mm\leq c\leq40mm, 10mm\leq d\leq80mm\\ &~&200\leq n_{1}\leq 1200, 0.15mm^{2}\leq S_{1}\leq19mm^{2}\\ &~&0.15mm^{2}\leq S_{2}\leq 19mm^{2} \end{array} $$
(C.14)

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Mirjalili, S., Jangir, P. & Saremi, S. Multi-objective ant lion optimizer: a multi-objective optimization algorithm for solving engineering problems. Appl Intell 46, 79–95 (2017). https://doi.org/10.1007/s10489-016-0825-8

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