HMOSHSSA: a hybrid meta-heuristic approach for solving constrained optimization problems

Abstract

This paper proposes a novel hybrid multi-objective optimization algorithm named HMOSHSSA by synthesizing the strengths of Multi-objective Spotted Hyena Optimizer (MOSHO) and Salp Swarm Algorithm (SSA). HMOSHSSA utilizes the exploration capability of MOSHO to explore the search space effectively and leader and follower selection mechanism of SSA to achieve global best solution with faster convergence. The proposed algorithm is evaluated on 24 benchmark test functions, and its performance is compared with seven well-known multi-objective optimization algorithms. The experimental results demonstrate that HMOSHSSA acquires very competitive results and outperforms other algorithms in terms of convergence speed, search-ability and accuracy. Additionally, HMOSHSSA is also applied on seven well-known engineering problems to further verify its efficacy. The results reveal the effectiveness of proposed algorithm toward solving real-life multi-objective optimization problems.

Appendices

Appendix 1: Unconstrained multi-objective test problems

Name	Mathematical formulation	Properties
UF1	$f_1=x_1+\dfrac{2}{\mid J_1 \mid }\sum _{j\epsilon J_1}[x_j-sin\Big (6\pi x_1+\dfrac{j\pi }{n}\Big )]^2$	Bi-objective
	$f_2=1-\sqrt{x}+\dfrac{2}{\mid J_2 \mid }\sum _{j\epsilon J_2}[x_j-sin\Big (6\pi x_1+\dfrac{j\pi }{n}\Big )]^2$
	$J_1=\{j\, \mid \, j \,\, is\,\, odd\,\, and\,\, 2 \le j \le n\} , J_2=\{j\, \mid \, j \,\, is\,\, even\,\, and\,\, 2 \le j \le n\}$
UF2	$f_1=x_1+\dfrac{2}{\mid J_1 \mid }\sum _{j\epsilon J_1}y_j^2$	Bi-objective
	$f_2=1-\sqrt{x}+\dfrac{2}{\mid J_2 \mid }\sum _{j\epsilon J_2}y_j^2$
	$J_1=\{j\, \mid \, j \,\, is\,\, odd\,\, and\,\, 2 \le j \le n\} , J_2=\{j\, \mid \, j \,\, is\,\, even\,\, and\,\, 2 \le j \le n\}$
	$y_j= {\left\{ \begin{array}{ll} x_j-[0.3x_1^2cos\Big (24\pi x_1+\dfrac{4j\pi }{n}\Big )+0.6x_1]cos\Big (6\pi x_1 + \dfrac{j\pi }{n}\Big ),&{} if \,\, j \epsilon J_1\\ x_j-[0.3x_1^2cos\Big (24\pi x_1+\dfrac{4j\pi }{n}\Big )+0.6x_1]sin\Big (6\pi x_1 + \dfrac{j\pi }{n}\Big ),&{} if \,\, j \epsilon J_2 \end{array}\right. }$
UF3	$f_1=x_1+\dfrac{2}{\mid J_1 \mid }\Big (4\sum _{j\epsilon J_1}y_j^2 - 2 \prod _{j \epsilon J_1}cos\Big (\dfrac{20y_j\pi }{\sqrt{j}} \Big )+2\Big )$	Bi-objective
	$f_2=\sqrt{x_1}+\dfrac{2}{\mid J_2 \mid }\Big (4\sum _{j\epsilon J_1}y_j^2 - 2 \prod _{j \epsilon J_2}cos\Big (\dfrac{20y_j\pi }{\sqrt{j}} \Big )+2\Big )$
	$J_1=\{j\, \mid \, j \,\, is\,\, odd\,\, and\,\, 2 \le j \le n\} , J_2=\{j\, \mid \, j \,\, is\,\, even\,\, and\,\, 2 \le j \le n\}$
	$y_j=x_j-x_1^{0.5(1.0+ \tfrac{3(j-2)}{n-2})} , j=2, 3, \ldots , n$
UF4	$f_1=x_1+\dfrac{2}{\mid J_1 \mid }\sum _{j\epsilon J_1}h(y_j)$	Bi-objective
	$f_2=1-x_2+\dfrac{2}{\mid J_2 \mid }\sum _{j\epsilon J_2}h(y_j)$
	$J_1=\{j\, \mid \, j \,\, is\,\, odd\,\, and\,\, 2 \le j \le n\} , J_2=\{j\, \mid \, j \,\, is\,\, even\,\, and\,\, 2 \le j \le n\}$
	$y_j=x_j-sin\Big (6\pi x_1 + \dfrac{j\pi }{n}\Big ) , j=2, 3, \ldots , n, \quad h(t)=\dfrac{\mid t \mid }{1+e^{2\mid t \mid }}$
	$f_1=x_1+\Big (\dfrac{1}{2N}+\epsilon \Big )\mid sin(2N\pi x_1)\mid +\dfrac{2}{\mid J_1 \mid }\sum _{j\epsilon J_1}h(y_i)$
UF5	$f_2=1-x_1+\Big (\dfrac{1}{2N}+\epsilon \Big ) \mid sin(2N\pi x_1) \mid + \dfrac{2}{\mid J_2 \mid }\sum _{j\epsilon J_2}h(y_i)$	Bi-objective
	$J_1=\{j\, \mid \, j \,\, is\,\, odd\,\, and\,\, 2 \le j \le n\} , J_2=\{j\, \mid \, j \,\, is\,\, even\,\, and\,\, 2 \le j \le n\}$
	$\epsilon > 0, \quad y_j=x_j-sin\Big (6\pi x_1 + \dfrac{j\pi }{n}\Big ) , j=2, 3, \ldots , n, \quad h(t)=2t^2-cos(4\pi t)+1$
UF6	$f_1=x_1+max\{0,2\Big (\dfrac{1}{2N}+\epsilon \Big )sin(2N\pi x_1)\}+\dfrac{2}{\mid J_1 \mid }\Big (4\sum _{j\epsilon J_1}y_j^2-2 \prod _{j\epsilon J_1} cos \Big (\dfrac{20y_j\pi }{\sqrt{j}} \Big )+1)$	Bi-objective
	$f_2=1-x_1+max\{0,2\Big (\dfrac{1}{2N}+\epsilon \Big )sin(2N\pi x_1)\}+\dfrac{2}{\mid J_2 \mid }\Big (4\sum _{j\epsilon J_2}y_j^2-2 \prod _{j\epsilon J_2} cos \Big (\dfrac{20y_j\pi }{\sqrt{j}} \Big )+1)$
	$J_1=\{j\, \mid \, j \,\, is\,\, odd\,\, and\,\, 2 \le j \le n\} , J_2=\{j\, \mid \, j \,\, is\,\, even\,\, and\,\, 2 \le j \le n\}$
	$\epsilon > 0, \quad y_j=x_j-sin\Big (6\pi x_1 + \dfrac{j\pi }{n}\Big ) , j=2, 3, \ldots , n$
UF7	$f_1= \root 5 \of {x_1} + \dfrac{2}{\mid J_1 \mid }\sum _{j\epsilon J_1}y_j^2$	Bi-objective
	$f_2=1-\root 5 \of {x_1} + \dfrac{2}{\mid J_2 \mid }\sum _{j\epsilon J_2}y_j^2$
	$J_1=\{j\, \mid \, j \,\, is\,\, odd\,\, and\,\, 2 \le j \le n\} , J_2=\{j\, \mid \, j \,\, is\,\, even\,\, and\,\, 2 \le j \le n\}$
	$\epsilon > 0, \quad y_j=x_j-sin\Big (6\pi x_1 + \dfrac{j\pi }{n}\Big ) , j=2, 3, \ldots , n$
UF8	$f_1=cos(0.5x_1\pi )cos(0.5x_2\pi )+\dfrac{2}{\mid J_1 \mid }\sum _{j\epsilon J_1}\Big (x_j-2x_2sin\Big (2\pi x_1+\dfrac{j\pi }{n}\Big )^2\Big )$	Tri-objective
	$f_2=cos(0.5x_1\pi )sin(0.5x_2\pi )+\dfrac{2}{\mid J_2 \mid }\sum _{j\epsilon J_2}\Big (x_j-2x_2sin\Big (2\pi x_1+\dfrac{j\pi }{n}\Big )^2\Big )$
	$f_3=sin(0.5x_1\pi )+\dfrac{2}{\mid J_3 \mid }\sum _{j\epsilon J_3}\Big (x_j-2x_2sin\Big (2\pi x_1+\dfrac{j\pi }{n}\Big )^2\Big )$
	$J_1=\{j\, \mid \, 3 \le j \le n, \,\, and \,\, j-1\,\, is\,\, a \,\, multiplication \,\, of \,\, 3\}$
	$J_2=\{j\, \mid \, 3 \le j \le n, \,\, and \,\, j-2\,\, is\,\, a \,\, multiplication \,\, of \,\, 3\}$
	$J_3=\{j\, \mid \, 3 \le j \le n, \,\, and \,\, j\,\, is\,\, a \,\, multiplication \,\, of \,\, 3\}$
UF9	$f_1=0.5[max\{0,(1+\epsilon )(1-4(2x_1-1)^2)\}+2x_1]x_2+\dfrac{2}{\mid J_1 \mid }\sum _{j\epsilon J_1}\Big (x_j-2x_2sin\Big (2\pi x_1+\dfrac{j\pi }{n}\Big )^2\Big )$	Tri-objective
	$f_2=0.5[max\{0,(1+\epsilon )(1-4(2x_1-1)^2)\}+2x_1]x_2+\dfrac{2}{\mid J_2 \mid }\sum _{j\epsilon J_2}\Big (x_j-2x_2sin\Big (2\pi x_1+\dfrac{j\pi }{n}\Big )^2\Big )$
	$f_3=1-x_2+\dfrac{2}{\mid J_3 \mid }\sum _{j\epsilon J_3}\Big (x_j-2x_2sin\Big (2\pi x_1+\dfrac{j\pi }{n}\Big )^2\Big )$
	$J_1=\{j\, \mid \, 3 \le j \le n, \,\, and \,\, j-1\,\, is\,\, a \,\, multiplication \,\, of \,\, 3\}$
	$J_2=\{j\, \mid \, 3 \le j \le n, \,\, and \,\, j-2\,\, is\,\, a \,\, multiplication \,\, of \,\, 3\}$
	$J_3=\{j\, \mid \, 3 \le j \le n, \,\, and \,\, j\,\, is\,\, a \,\, multiplication \,\, of \,\, 3\},\quad \epsilon =0.1$
UF10	$f_1=cos(0.5x_1\pi )cos(0.5x_2\pi )+\dfrac{2}{\mid J_1 \mid }\sum _{j\epsilon J_1}[4y_j^2-cos(8\pi y_j)+1]$	Tri-objective
	$f_2=cos(0.5x_1\pi )sin(0.5x_2\pi )+\dfrac{2}{\mid J_2 \mid }\sum _{j\epsilon J_2}[4y_j^2-cos(8\pi y_j)+1]$
	$f_3=sin(0.5x_1\pi )+\dfrac{2}{\mid J_3 \mid }\sum _{j\epsilon J_3}[4y_j^2-cos(8\pi y_j)+1]$
	$J_1=\{j\, \mid \, 3 \le j \le n, \,\, and \,\, j-1\,\, is\,\, a \,\, multiplication \,\, of \,\, 3\}$
	$J_2=\{j\, \mid \, 3 \le j \le n, \,\, and \,\, j-2\,\, is\,\, a \,\, multiplication \,\, of \,\, 3\}$
	$J_3=\{j\, \mid \, 3 \le j \le n, \,\, and \,\, j\,\, is\,\, a \,\, multiplication \,\, of \,\, 3\}$

Appendix 2: Unconstrained multi-objective test problems

ZDT1: Convex

$$\begin{aligned}&Minimize:\quad f_1(x)=x_1\\&Minimize:\quad f_2(x)=g(x)\times h(f_1(x),g(x))\\ \end{aligned}$$

where,

$$\begin{aligned}&g(x)=1+\dfrac{9}{N-1}\mathop \sum \limits _{i=2}^{N}x_i\\&h(f_1(x),g(x))=1-\sqrt{\dfrac{f_1(x)}{g(x)}}\\&0\le x_i\le 1,\, 1\le i \le 30 \end{aligned}$$

ZDT2: Concave

$$\begin{aligned}&Minimize:\quad f_1(x)=x_1\\&Minimize:\quad f_2(x)=g(x)\times h(f_1(x),g(x))\\ \end{aligned}$$

where,

$$\begin{aligned}&g(x)=1+\dfrac{9}{N-1}\mathop \sum \limits _{i=2}^{N}x_i\\&h(f_1(x),g(x))=1-\Bigg (\dfrac{f_1(x)}{g(x)}\Bigg )^2\\&0\le x_i\le 1,\, 1\le i \le 30\\ \end{aligned}$$

ZDT3: Disconnected

$$\begin{aligned}&Minimize:\quad f_1(x)=x_1\\&Minimize:\quad f_2(x)=g(x)\times h(f_1(x),g(x))\\ \end{aligned}$$

where,

$$\begin{aligned}&g(x)=1+\dfrac{9}{29}\mathop \sum \limits _{i=2}^{N}x_i\\&h(f_1(x),g(x))=1-\sqrt{\dfrac{f_1(x)}{g(x)}}\\&\quad -\Bigg (\dfrac{f_1(x)}{g(x)}\Bigg )sin(10 \pi f_1(x))\\&0\le x_i\le 1,\, 1\le i \le 30\\ \end{aligned}$$

ZDT4: Convex

$$\begin{aligned}&Minimize: f_1(x)=x_1\\&Minimize: f_2(x)=g(x)\times [1-(x_1/g(x))^2]\\ \end{aligned}$$

where,

$$\begin{aligned}&g(x)=1+10(n-1)+\mathop \sum \limits _{i=2}^{n}(x_i^2-10cos(4\pi x_i))\\&0\le x_1\le 1,\, -5\le x_i \le 5,\, i=1,2,\ldots ,n\\ \end{aligned}$$

ZDT6: Concave

$$\begin{aligned}&Minimize:\quad f_1(x)=1-e^{-4x_1}\times sin^6(6\pi x_1)\\&Minimize:\quad f_2(x)=1-\Big (\dfrac{f_1(x)}{g(x)} \Big )^2\\ \end{aligned}$$

where,

$$\begin{aligned}&g(x)=1+9\Bigg [\dfrac{\Big (\sum _{i=2}^{n}x_i \Big )}{(n-1)} \Bigg ]^{0.25}\\&0\le x_i\le 1,\, i=1,2,\ldots ,n \end{aligned}$$

Appendix 3: Unconstrained multi-objective test problems

DTLZ1: Linear

$$\begin{aligned}&Minimize:\quad f_1(\vec {x})=\dfrac{1}{2}x_1(1+g(\vec {x}))\\&Minimize:\quad f_2(\vec {x})=\dfrac{1}{2}(1-x_1)(1+g(\vec {x}))\\ \end{aligned}$$

where,

$$\begin{aligned}&g(\vec {x})=100\Big [\mid \vec {x} \mid +\mathop \sum \limits _{x_i\epsilon \vec {x}}(x_1-0.5)^2-cos(20\pi (x_i-0.5))\Big ]\\&0\le x_i\le 1,\, i=1,2,\ldots ,n\\ \end{aligned}$$

DTLZ2: Concave

$$\begin{aligned}&Minimize:\quad f_1(\vec {x})=(1+g(\vec {x}))cos\Big (x_1\dfrac{\pi }{2}\Big )\\&Minimize:\quad f_2(\vec {x})=(1+g(\vec {x}))sin\Big (x_1\dfrac{\pi }{2}\Big )\\ \end{aligned}$$

where,

$$\begin{aligned}&g(\vec {x})=\mathop \sum \limits _{x_i\epsilon \vec {x}}(x_i-0.5)^2\\&0\le x_i\le 1,\, i=1,2,\ldots ,n\\ \end{aligned}$$

DTLZ3: Concave

$$\begin{aligned} Minimize:\quad f_1(\vec {x})=(1+g(\vec {x}))cos\Big (x_1\dfrac{\pi }{2}\Big )\\ Minimize:\quad f_2(\vec {x})=(1+g(\vec {x}))sin\Big (x_1\dfrac{\pi }{2}\Big )\\ \end{aligned}$$

where,

$$\begin{aligned}&g(\vec {x})=100\Big [\mid \vec {x} \mid + \mathop \sum \limits _{x_i\epsilon \vec {x}}(x_i-0.5)^2-cos(20\pi (x_i-0.5))\Big ]\\&0\le x_i\le 1,\, i=1,2,\ldots ,n\\ \end{aligned}$$

DTLZ4: Concave

$$\begin{aligned}&Minimize:\quad f_1(\vec {x})=(1+g(\vec {x}))cos\Big (x_1^\alpha \dfrac{\pi }{2}\Big )\\&Minimize:\quad f_2(\vec {x})=(1+g(\vec {x}))sin\Big (x_1^\alpha \dfrac{\pi }{2}\Big )\\ \end{aligned}$$

where,

$$\begin{aligned}&g(\vec {x})=\mathop \sum \limits _{x_i\epsilon \vec {x}}(x_i-0.5)^2\\&\alpha =100\\&0\le x_i\le 1,\, i=1,2,\ldots ,n\\ \end{aligned}$$

DTLZ5:

$$\begin{aligned}&Minimize:\quad f_1(\vec {x})=(1+g(\vec {x}))cos\Big (\dfrac{1+2g(\vec {x})x_1}{4(1+g(\vec {x}))}\times \dfrac{\pi }{2}\Big )\\&Minimize:\quad f_2(\vec {x})=(1+g(\vec {x}))sin\Big (\dfrac{1+2g(\vec {x})x_1}{4(1+g(\vec {x}))}\times \dfrac{\pi }{2}\Big )\\ \end{aligned}$$

where,

$$\begin{aligned}&g(\vec {x})=\mathop \sum \limits _{x_i\epsilon \vec {x}}(x_i-0.5)^2\\&0\le x_i\le 1,\, i=2, 3, \ldots , n\\ \end{aligned}$$

DTLZ6:

$$\begin{aligned}&Minimize:\quad f_1(\vec {x})=(1+g(\vec {x}))cos\Big (\dfrac{1+2g(\vec {x})x_1}{4(1+g(\vec {x}))}\times \dfrac{\pi }{2}\Big )\\&Minimize:\quad f_2(\vec {x})=(1+g(\vec {x}))sin\Big (\dfrac{1+2g(\vec {x})x_1}{4(1+g(\vec {x}))}\times \dfrac{\pi }{2}\Big )\\ \end{aligned}$$

where,

$$\begin{aligned}&g(\vec {x})=\mathop \sum \limits _{x_i\epsilon \vec {x}}x_i^{0.1}\\&0\le x_i\le 1,\, i=2, 3, \ldots , n\\ \end{aligned}$$

DTLZ7: Disconnected

$$\begin{aligned}&Minimize:\quad f_1(\vec {x})=x_1\\&Minimize:\quad f_2(\vec {x})=(1+g(\vec {x}))h(f_1(\vec {x}),g(\vec {x}))\\ \end{aligned}$$

where,

$$\begin{aligned}&g(\vec {x})=1+\dfrac{9}{\mid \vec {x} \mid }\mathop \sum \limits _{x_i\epsilon \vec {x}}x_i\\&h(f_1(\vec {x}),g(\vec {x}))=M-\dfrac{f_1(\vec {x})}{1+g(\vec {x})}(1+sin(3\pi f_1(\vec {x})))\\&0\le x_i\le 1,\, 1\le i \le n \end{aligned}$$

Appendix 4: Constrained engineering design problems

1.1 Welded beam design problem

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

1.2 Multiple-disk clutch brake design problem

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

1.3 Pressure vessel design problem

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

1.4 Speed reducer design problem

$$\begin{aligned}&{\text{Minimize}}\,\, f_1(\vec {K}) = \dfrac{\sqrt{ \dfrac{745z_4}{z_2z_3}^2 + 1.69 \times 10^7}}{0.1z_6^3},\\&{\text{Minimize}},\, f_2(\vec {K}) = 0.7854z_1z_2^2(3.3333z_3^2 + 14.9334z_3 - 43.0934)\\ {}&- 1.508z_1(z_6^2 + z_7^2) + 7.4777(z_6^3 + z_7^3) + 0.7854(z_4z_6^2 + z_5z_7^2),\\&{\text{Subject}}\;{\text{ to}}\\&g_1(\vec {K}) = \dfrac{27}{z_1z_2^2z_3} - 1 \le 0,\\&g_2(\vec {K}) = \dfrac{397.5}{z_1z_2^2z_3^2} - 1 \le 0,\\&g_3(\vec {K}) = \dfrac{1.93z_4^3}{z_2z_6^4z_3} - 1 \le 0,\\&g_4(\vec {K}) = \dfrac{1.93z_5^3}{z_2z_7^4z_3} - 1 \le 0,\\&g_5(\vec {K}) = \dfrac{[(745(z_4/z_2z_3))^2 + 16.9 \times 10^6]^{1/2}}{110z_6^3} - 1 \le 0,\\&g_6(\vec {K}) = \dfrac{[(745(z_5/z_2z_3))^2 + 157.5 \times 10^6]^{1/2}}{85z_7^3} - 1 \le 0,\\&g_7(\vec {K}) = \dfrac{z_2z_3}{40} - 1 \le 0,\\&g_8(\vec {K}) = \dfrac{5z_2}{z_1} - 1 \le 0,\\&g_9(\vec {K}) = \dfrac{z_1}{12z_2} - 1 \le 0,\\&g_{10}(\vec {K}) = \dfrac{1.5z_6 + 1.9}{z_4} - 1 \le 0,\\&g_{11}(\vec {K}) = \dfrac{1.1z_7 + 1.9}{z_5} - 1 \le 0,\\&where\\&2.6 \le z_1 \le 3.6, \,\,0.7\le z_2 \le 0.8, \,\,17 \le z_3 \le 28,\,\,7.3 \le z_4 \le 8.3,\\&7.3 \le z_5 \le 8.3, \,\,2.9 \le z_6 \le 3.9, \,\,5.0 \le z_7 \le 5.5 \,\, \end{aligned}$$

1.5 Gear train design problem

$$\begin{aligned}&{\text{Minimize}}\,\, f_1(T) = \Big | 6.931 - \dfrac{T_a}{T_d} \dfrac{T_f}{T_b}\Big |\\&{\text{Minimize}}\,\, f_2(T) = max \Big (T_d, T_b, T_a, T_f\Big )\\&{\text{Subject}}\;{\text{to}}\\&\dfrac{f_1(T)}{6.931} \le 0.5,\\&12 \le T_d, T_b, T_a, T_f \le 60,\\&T_d, T_b, T_a, T_f {\text{ are}}\;{\text{ integers}} \end{aligned}$$