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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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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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