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EMoSOA: a new evolutionary multi-objective seagull optimization algorithm for global optimization

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Abstract

This study introduces the evolutionary multi-objective version of seagull optimization algorithm (SOA), entitled Evolutionary Multi-objective Seagull Optimization Algorithm (EMoSOA). In this algorithm, a dynamic archive concept, grid mechanism, leader selection, and genetic operators are employed with the capability to cache the solutions from the non-dominated Pareto. The roulette-wheel method is employed to find the appropriate archived solutions. The proposed algorithm is tested and compared with state-of-the-art metaheuristic algorithms over twenty-four standard benchmark test functions. Four real-world engineering design problems are validated using proposed EMoSOA algorithm to determine its adequacy. The findings of empirical research indicate that the proposed algorithm is better than other algorithms. It also takes into account those optimal solutions from the Pareto which shows high convergence.

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Notes

  1. The term fitness value is defined as a process which evaluates the population and gives a score or fitness. Whereas, the process is a function which measures the quality of the represented solution.

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Acknowledgements

This work is partly supported by VC Research (VCR 0000056) for Prof Chang.

Author information

Authors and Affiliations

  1. Department of Computer Science, Government Bikram College of Commerce, Patiala, Punjab, 147001, India

    Gaurav Dhiman

  2. Department of Electronics and Communication Engineering, KIET Group of Institution, Delhi-NCR, Ghaziabad, India

    Krishna Kant Singh

  3. Department of Electronics and Computer Science, Koszalin University of Technology, Sniadeckich 2, 75-453, Koszalin, Poland

    Adam Slowik

  4. School of Computing, Engineering and Digital Technologies, Teesside University, Middlesbrough, UK

    Victor Chang

  5. Department of Automotive Engineering, College of Engineering, Uludag University, Grkle, Bursa, 16059, Turkey

    Ali Riza Yildiz

  6. Department of Computer Science and Engineering, Sri Guru Granth Sahib World University, Fatehgarh Sahib, Punjab, India

    Amandeep Kaur

  7. Department of Computer Science, Government Bikram College of Commerce, Patiala, Punjab, 147001, India

    Meenakshi Garg

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  1. Gaurav Dhiman
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  2. Krishna Kant Singh
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  4. Victor Chang
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  7. Meenakshi Garg
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The source codes are available at: http://dhimangaurav.com/.

Appendices

Appendix A: Unconstrained multi-objective test problems

See Table 12.

Table 12 Table caption
Full size table

Appendix B: Unconstrained multi-objective test problems

  • ZDT1:

    $$\begin{aligned} \begin{array}{ll} \text {Minimize}:&{}\quad f_1(x)=x_1\\ \text {Minimize}:&{}\quad f_2(x)=g(x)\times h(f_1(x),g(x))\\ where,&{}\\ &{} \quad g(x)=1+\dfrac{9}{N-1}{\sum }_{i=2}^{N}x_i\\ &{}\quad h(f_1(x),g(x))=1-\sqrt{\dfrac{f_1(x)}{g(x)}}\\ &{}\quad 0\le x_i\le 1,\, 1\le i \le 30\ \end{array} \end{aligned}$$

  • ZDT2:

    $$\begin{aligned} \begin{array}{ll} \text {Minimize}:&{}\quad f_1(x)=x_1\\ \text {Minimize}:&{}\quad f_2(x)=g(x)\times h(f_1(x),g(x))\\ where,&{}\\ &{}\quad g(x)=1+\dfrac{9}{N-1}{\sum }_{i=2}^{N}x_i\\ &{}\quad h(f_1(x),g(x))=1-\Bigg (\dfrac{f_1(x)}{g(x)}\Bigg )^2\\ &{}\quad 0\le x_i\le 1,\, 1\le i \le 30\\ \end{array} \end{aligned}$$

  • ZDT3:

    $$\begin{aligned} \begin{array}{ll} \text {Minimize}:&{}\quad f_1(x)=x_1\\ \text {Minimize}:&{}\quad f_2(x)=g(x)\times h(f_1(x),g(x))\\ where,&{}\\ &{}\quad g(x)=1+\dfrac{9}{29}{\sum }_{i=2}^{N}x_i\\ &{}\quad h(f_1(x),g(x))=1-\sqrt{\dfrac{f_1(x)}{g(x)}}\\ &{}\quad -\Bigg (\dfrac{f_1(x)}{g(x)}\Bigg )sin(10 \pi f_1(x))\\ &{}\quad 0\le x_i\le 1,\, 1\le i \le 30\\ \end{array} \end{aligned}$$

  • ZDT4:

    $$\begin{aligned} \begin{array}{ll} \text {Minimize}:&{}\quad f_1(x)=x_1\\ \text {Minimize}:&{}\quad f_2(x)=g(x)\times [1-(x_1/g(x))^2]\\ where,&{}\\ &{}\quad g(x)=1+10(n-1)+{\sum }_{i=2}^{n}(x_i^2-10cos(4\pi x_i))\\ &{}\quad 0\le x_1\le 1,\, -5\le x_i \le 5,\, i=1,2,\ldots ,n\ \end{array} \end{aligned}$$

  • ZDT6:

    $$\begin{aligned} \begin{array}{ll} \text {Minimize}:&{}\quad f_1(x)=1-e^{-4x_1}\times sin^6(6\pi x_1)\\ \text {Minimize}:&{}\quad f_2(x)=1-\Big (\dfrac{f_1(x)}{g(x)} \Big )^2\\ where,&{}\\ &{}\quad g(x)=1+9\Bigg [\dfrac{\Big (\sum _{i=2}^{n}x_i \Big )}{(n-1)} \Bigg ]^{0.25}\\ &{}\quad 0\le x_i\le 1,\, i=1,2,\ldots ,n\\ \end{array} \end{aligned}$$

Appendix C: Unconstrained multi-objective test problems

  • DTLZ1:

    $$\begin{aligned} \begin{array}{ll} \text {Minimize}:&{}\quad f_1(\vec {x})=\dfrac{1}{2}x_1(1+g(\vec {x}))\\ \text {Minimize}:&{}\quad f_2(\vec {x})=\dfrac{1}{2}(1-x_1)(1+g(\vec {x}))\\ where,&{}\\ &{}\quad g(\vec {x})=100\Big [\mid \vec {x} \mid +{\sum }_{x_i\epsilon \vec {x}}(x_1-0.5)^2-cos(20\pi (x_i-0.5))\Big ]\\ &{}\quad 0\le x_i\le 1,\, i=1,2,\ldots ,n\\ \end{array} \end{aligned}$$

  • DTLZ2:

    $$\begin{aligned} \begin{array}{ll} \text {Minimize}:&{}\quad f_1(\vec {x})=(1+g(\vec {x}))cos\Big (x_1\dfrac{\pi }{2}\Big )\\ \text {Minimize}:&{}\quad f_2(\vec {x})=(1+g(\vec {x}))sin\Big (x_1\dfrac{\pi }{2}\Big )\\ where,&{}\\ &{}\quad g(\vec {x})={\sum }_{x_i\epsilon \vec {x}}(x_i-0.5)^2\\ &{}\quad 0\le x_i\le 1,\, i=1,2,\ldots ,n\\ \end{array} \end{aligned}$$

  • DTLZ3:

    $$\begin{aligned} \begin{array}{ll} \text {Minimize}:&{}\quad f_1(\vec {x})=(1+g(\vec {x}))cos\Big (x_1\dfrac{\pi }{2}\Big )\\ \text {Minimize}:&{}\quad f_2(\vec {x})=(1+g(\vec {x}))sin\Big (x_1\dfrac{\pi }{2}\Big )\\ where,&{}\\ &{}\quad g(\vec {x})=100\Big [\mid \vec {x} \mid + {\sum }_{x_i\epsilon \vec {x}}(x_i-0.5)^2-cos(20\pi (x_i-0.5))\Big ]\\ &{}\quad 0\le x_i\le 1,\, i=1,2,\ldots ,n\\ \end{array} \end{aligned}$$

  • DTLZ4:

    $$\begin{aligned} \begin{array}{ll} \text {Minimize}:&{}\quad f_1(\vec {x})=(1+g(\vec {x}))cos\Big (x_1^\alpha \dfrac{\pi }{2}\Big )\\ \text {Minimize}:&{}\quad f_2(\vec {x})=(1+g(\vec {x}))sin\Big (x_1^\alpha \dfrac{\pi }{2}\Big )\\ where,&{}\\ &{}\quad g(\vec {x})={\sum }_{x_i\epsilon \vec {x}}(x_i-0.5)^2\\ &{}\quad \alpha =100\\ &{}\quad 0\le x_i\le 1,\, i=1,2,\ldots ,n\\ \end{array} \end{aligned}$$

  • DTLZ5:

    $$\begin{aligned} \begin{array}{ll} \text {Minimize}:&{}\quad f_1(\vec {x})=(1+g(\vec {x}))cos\Big (\dfrac{1+2g(\vec {x})x_1}{4(1+g(\vec {x}))}\times \dfrac{\pi }{2}\Big )\\ \text {Minimize}:&{}\quad f_2(\vec {x})=(1+g(\vec {x}))sin\Big (\dfrac{1+2g(\vec {x})x_1}{4(1+g(\vec {x}))}\times \dfrac{\pi }{2}\Big )\\ where,&{}\\ &{}\quad g(\vec {x})={\sum }_{x_i\epsilon \vec {x}}(x_i-0.5)^2\\ &{}\quad 0\le x_i\le 1,\, i=2, 3, \ldots , n\\ \end{array} \end{aligned}$$

  • DTLZ6:

    $$\begin{aligned} \begin{array}{ll} \text {Minimize}:&{}\quad f_1(\vec {x})=(1+g(\vec {x}))cos\Big (\dfrac{1+2g(\vec {x})x_1}{4(1+g(\vec {x}))}\times \dfrac{\pi }{2}\Big )\\ \text {Minimize}:&{}\quad f_2(\vec {x})=(1+g(\vec {x}))sin\Big (\dfrac{1+2g(\vec {x})x_1}{4(1+g(\vec {x}))}\times \dfrac{\pi }{2}\Big )\\ where,&{}\\ &{}\quad g(\vec {x})={\sum }_{x_i\epsilon \vec {x}}x_i^{0.1}\\ &{}\quad 0\le x_i\le 1,\, i=2, 3, \ldots , n\\ \end{array} \end{aligned}$$

  • DTLZ7:

    $$\begin{aligned} \begin{array}{ll} \text {Minimize}:&{}\quad f_1(\vec {x})=x_1\\ \text {Minimize}:&{}\quad f_2(\vec {x})=(1+g(\vec {x}))h(f_1(\vec {x}),g(\vec {x}))\\ where,&{}\\ &{}\quad g(\vec {x})=1+\dfrac{9}{\mid \vec {x} \mid }{\sum }_{x_i\epsilon \vec {x}}x_i\\ &{}\quad h(f_1(\vec {x}),g(\vec {x}))=M-\dfrac{f_1(\vec {x})}{1+g(\vec {x})}(1+sin(3\pi f_1(\vec {x})))\\ &{}\quad 0\le x_i\le 1,\, 1\le i \le n\\ \end{array} \end{aligned}$$

Appendix D: Constrained engineering design problems

1.1 D.1. Welded beam design problem

$$\begin{aligned} \begin{aligned} \begin{aligned}&\text {Minimize}\,\, f_1(\vec {z}) = C = 1.10471h^2l + 0.04811tb(14.0 + l),\\&\text {Minimize}\,\, f_2(\vec {z}) = D = \dfrac{2.1952}{t^3b},\\&\text {Subject to}\\&g_1(\vec {z}) = 13,600 - \tau {(\vec {z})} \ge 0,\\&g_2(\vec {z}) = 30,000 - \sigma {(\vec {z})} \ge 0,\\&g_3(\vec {z}) = b - h \ge 0,\\&g_4(\vec {z}) = P_c(\vec {z}) - 6,000 \ge 0,\\ \end{aligned} \end{aligned}\\ \begin{aligned} \text {Variable range} \\&0.125 \le h, b \le 5.0 \text {in.},\\&0.1 \le l, t \le 10.0 \text {in.},\\ \end{aligned}\\ \begin{aligned} \text {where}\\&\tau {(\vec {z})} = \sqrt{(\tau ^{'})^2 + (\tau ^{''})^2 + (l\tau ^{'}\tau ^{''})/\sqrt{0.25(l^2 + (h + t)^2)}},\\&\tau ^{'} = \dfrac{6,000}{\sqrt{2}hl}, \sigma {(\vec {z})} = \dfrac{504,000}{t^2b},\\&\tau ^{''} = \dfrac{6,000(14 + 0.5l)\sqrt{0.25(l^2 + (h + t)^2)}}{2[0.707hl(l^2/12 + 0.25(h + t)^2)]},\\&P_c(\vec {z}) = 64,746.022(1 - 0.0282346t)tb^3. \end{aligned} \end{aligned}$$

D.2. Multiple-disk clutch brake design problem

$$\begin{aligned} \begin{aligned} \begin{aligned}&\text {Minimize}\,\, f_1(\vec {z}) = M = \pi (r_o^2 - r_i^2)t(Z+1)p_m,\\&\text {Minimize}\,\, f_2(\vec {z}) = T = \dfrac{I_zw}{M_h+M_f},\\&\text {Subject to} \\&g_1(\vec {z}) = r_o - r_i - \Delta R \ge 0,\\&g_2(\vec {z}) = L_{max} - (Z+1)(t+\delta ) \ge 0,\\&g_3(\vec {z}) = p_{max} - p_{rz} \ge 0,\\&g_4(\vec {z}) = p_{max}V_{sr,max} - p_{rz}V_{sr} \ge 0,\\&g_5(\vec {z}) = V_{sr,max} - V_{sr} \ge 0,\\&g_6(\vec {z}) = M_h - sM_s \ge 0,\\&g_7(\vec {z}) = T \ge 0,\\&g_8(\vec {z}) = T_{max} - T \ge 0,\\&60 \le r_i \le 80\text { mm},\\&90 \le r_o \le 110\text { mm},\\&1.5 \le t \le 3\text { mm},\\&0 \le F \le 1000\text { N},\\&2 \le Z\le 9\\ \text {where}\\&p_m = 0.0000078 \text { kg/mm}^3, p_{max} = 1 \text { MPa}, \mu = 0.5, V_{sr,max} = 10 \text { m/s},\\&s = 1.5, T_{max} = 15\text { s}, n = 250 \text { rpm}, M_s = 40 \text { Nm}, M_f = 3 \text { Nm},\\&I_z = 55 \text { kg-m}^2, \delta = 0.5 \text { mm}, \Delta R = 20 \text { mm}, L_{max} = 30 \text { mm},\\&M_h = \dfrac{2}{3}\mu FZ\dfrac{r_o^3 - r_i^3}{r_o^2 - r_i^2} \text { N-mm}, w = \dfrac{\pi n}{30} \text { rad/s}, R_{sr} = \dfrac{2}{3}\dfrac{r_o^3 - r_i^3}{r_o^2 - r_i^2} \text { mm}\\&A = \pi (r_o^2 - r_i^2) \text { mm}^2, p_{rz} = \dfrac{F}{A} \text { N/mm}^2, V_{sr} = \dfrac{\pi R_{sr}n}{30} \text { mm/s}, \end{aligned} \end{aligned} \end{aligned}$$

D.3. Pressure vessel design problem

$$\begin{aligned} \begin{aligned} \begin{aligned}&\text {Minimize}\,\, f_1(\vec {z}) = 0.6224T_sLR + 1.7781T_hR^2 + 3.1661T_s^2L + 19.84T_s^2R,\\&\text {Minimize}\,\, f_2(\vec {z}) = -(\pi R^2L + 1.333\pi R^3),\\&\text {Subject to}\\&g_1(\vec {z}) = 0.0193R - T_s \le 0,\\&g_2(\vec {z}) = 0.00954R - T_h \le 0,\\&g_3(\vec {z}) = 0.0625 - T_s \le 0,\\&g_4(\vec {z}) = T_s - 5 \le 0,\\&g_5(\vec {z}) = 0.0625 - T_h \le 0,\\&g_6(\vec {z}) = T_h - 5 \le 0,\\&g_7(\vec {z}) = 10 - R \le 0,\\&g_8(\vec {z}) = R - 200 \le 0,\\&g_9(\vec {z}) = 10 - L \le 0,\\&g_{10}(\vec {z}) = L - 240 \le 0,\\ \end{aligned} \end{aligned} \end{aligned}$$

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Dhiman, G., Singh, K.K., Slowik, A. et al. EMoSOA: a new evolutionary multi-objective seagull optimization algorithm for global optimization. Int. J. Mach. Learn. & Cyber. 12, 571–596 (2021). https://doi.org/10.1007/s13042-020-01189-1

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