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

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.

Appendices

Appendix A: Unconstrained multi-objective test problems

See Table 12.

Table 12 Table caption

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

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