Appendix A: constrained multi-objective test problems used in this paper
CONSTR
This issue [47] contains two constraints and two design variables that have a convex Pareto front.
$${\text{Minimize: }} f_{1} \left( x \right) = x_{1}$$
(A.1)
$${\text{Minimize:}}\quad f_{2} \left( x \right) = \left( {1 + x_{2} } \right)/x_{1}$$
(A.2)
$${\text{where}}\quad g_{1} \left( x \right) = 6 - \left( {x_{2} + 9x_{1} } \right)$$
(A.3)
$$g_{2} \left( x \right) = 1 + \left( {x_{2} - 9x_{1} } \right)$$
(A.4)
$$0.1 \le x_{1} \le 1, 0 \le x_{2} \le 5$$
SRN
Srinivas and Deb [48] proposed the following continuous Pareto optimum front for the following issue:
$${\text{Minimize: }}\quad f_{1} \left( x \right) = 2 + \left( {x_{1} {-} 2} \right)^{2} + \left( {x_{2} {-} 1} \right)^{2}$$
(A.5)
$${\text{Minimize:}}\quad f_{2} \left( x \right){ } = 9x_{1} {-} \left( {x_{2} {-} 1} \right)^{2}$$
(A.6)
$${\text{where}}\quad g_{1} \left( x \right){ } = x_{1}^{2} + x_{2}^{2} - 255$$
(A.7)
$$g_{2} \left( x \right) = x_{1} - 3x_{2} + 10$$
(A.8)
$$- 20 \le x_{1} \le 20, - 20 \le x_{2} \le 20$$
BNH
Binh and Korn [49] provided a this example for the first time as follows:
$${\text{Minimize:}}\quad f_{1} \left( x \right) = 4x_{1}^{2} + 4x_{2}^{2}$$
(A.9)
$${\text{Minimize:}}\quad f_{2} \left( x \right) = (x_{1} - 5)^{2} + (x_{2} - 5)^{2}$$
(A.10)
$${\text{where}}\quad g_{1} \left( x \right) = (x_{1} - 5)^{2} + x_{2}^{2} - 25$$
(A.11)
$$g_{2} \left( x \right) = 7.7 - (x_{1} - 8)^{2} - (x_{2} + 3)^{2}$$
(A.12)
$$0 \le x_{1} \le 5,0 \le x_{2} \le 3$$
OSY
It was suggested by Osyczka and Kundu [50] that there should be five separate regions. In addition, the following six restrictions and six design variables should be taken into account:
$${\text{Minimize}}:\quad f_{1} \left( x \right) = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} + x_{5}^{2} + x_{6}^{2}$$
(A.13)
$${\text{Minimize}}:\quad f_{2} \left( x \right) = [25(x_{1} - 2)^{2} + (x_{2} - 1)^{2} + \left( {x_{3} - 1} \right) + (x_{4} - 4)^{2} + \left( {x_{5} - 1)^{2} } \right]$$
(A.14)
$${\text{where}} g_{1} \left( x \right) = 2 - x_{1} - x_{2}$$
(A.15)
$$g_{2} \left( x \right) = - 6 + x_{1} + x_{2}$$
(A.16)
$$g_{3} \left( x \right) = - 2 - x_{1} + x_{2}$$
(A.17)
$$g_{4} \left( x \right) = - 2 + x_{1} - 3x_{2}$$
(A.18)
$$g_{5} \left( x \right) = - 4 + x_{4} + (x_{3} - 3)^{2}$$
(A.19)
$$g_{6} \left( x \right) = 4 - x_{6} - (x_{5} - 3)^{2}$$
(A.20)
$$0 \le x_{1} \le 10, 0 \le x_{2} \le 10, 1 \le x_{3} \le 5$$
(A.21)
$$0 \le x_{4} \le 6, 1 \le x_{5} \le 5, 0 \le x_{6} \le 10$$
Appendix B: constrained multi-objective engineering problems used in this paper
The four-bar truss design problem
In this 4-bar truss issue [51], there are two objectives as structural volume (\({f}_{1}\)) and displacement (\({f}_{2}\)), which are considered to be minimized. This issue has four design variables (\({x}_{1}-{x}_{4}\)) according to the cross-sectional area of members 1, 2, 3, and 4. This problem may be mathematically formularized as below:
$${\text{Minimize: }} f_{1} \left( x \right) = 200 \times \left( {2 \times x\left( 1 \right) + {\text{sqrt}}\left( {2 \times x\left( 2 \right)} \right) + {\text{sqrt}}\left( {x\left( 3 \right)} \right) + x\left( 4 \right)} \right)$$
(B.1)
$${\text{Minimize: }} f_{2} \left( x \right) = 0.01 \times \left( {\frac{2}{x\left( 1 \right)}} \right) + \left( {\frac{{2 \times {\text{sqrt}}\left( 2 \right)}}{x\left( 2 \right)}} \right) - \left( {\left( {2 \times {\text{sqrt}}\left( 2 \right)} \right)/x\left( 3 \right)} \right) + \left( {2/x\left( 1 \right)} \right))$$
(B.2)
$$1 \le x_{1} \le 3, 1.4142 \le x_{2} \le 3$$
$$1.4142 \le x_{3} \le 3, 1 \le x_{4} \le 3$$
The welded beam design problem
Ray and Liew [52] first offered this welded beam issue with two objectives, namely the fabrication cost (\({f}_{1}\)) and beam deflection (\({f}_{2}\)), which is to be reduced and four constraints. This issue has four numbers of variables, named weld thickness (\({x}_{1}\)), clamped bar length (\({x}_{2}\)), bar height (\({x}_{3}\)), and bar thickness (\({x}_{4}\)) as follows:
$${\text{Minimize: }} f_{1} \left( x \right) = 1.10471 \times x\left( 1 \right)^{2} \times x\left( 2 \right) + 0.04811 \times x\left( 3 \right) \times x\left( 4 \right) \times \left( {14 + x\left( 2 \right)} \right)$$
(B.3)
$${\text{Minimize: }} f_{2} \left( x \right) = 65856000/\left( {30 \times 10^{6} \times x\left( 4 \right) \times x\left( 3 \right)^{3} } \right)$$
(B.4)
$${\text{where }} g_{1} \left( x \right) = \tau - 13600$$
(B.5)
$$g_{2} \left( x \right) = \sigma - 30000$$
(B.6)
$$g_{3} \left( x \right) = x\left( 1 \right) - x\left( 4 \right)$$
(B.7)
$$g_{4} \left( x \right) = 6000 - P$$
(B.8)
$$0.125 \le x_{1} \le 5, 0.1 \le x_{2} \le 10$$
$$0.1 \le x_{3} \le 10, 0.125 \le x_{4} \le 5$$
$${\text{where }} q = 6000*\left( {14 + \frac{x\left( 2 \right)}{2}} \right);\;D = {\text{sqrt}}\left( {\frac{{x(2)^{2} }}{4} + \frac{{x\left( 1 \right) + x\left( 3 \right))^{2} }}{4}} \right)$$
(B.9)
$$J = 2*\left( {x\left( 1 \right)*x\left( 2 \right)*{\text{sqrt}}\left( 2 \right)*\left( {\frac{{x(2)^{2} }}{12} + \frac{{(x\left( 1 \right) + x\left( 3 \right))^{2} }}{4}} \right)} \right)$$
(B.10)
$$\alpha = \frac{6000}{{{\text{sqrt}}\left( 2 \right)*x\left( 1 \right)*x\left( 2 \right)}}$$
(B.11)
$$\beta = Q*\frac{D}{J}$$
(B.12)
Disk brake design problem
This disk brake design issue has figured by Ray and Liew [52] with two objectives, namely stopping time (\({f}_{1}\)) and brake mass (\({f}_{2}\)) to minimize and five constraints for a disk brake. This problem contains five numbers of variables: disk inner radius (\({x}_{1}\)), outer disk radius (\({x}_{2}\)), engaging force (\({x}_{3}\)), and friction surfaces number (\({x}_{4}\)). The following equations may represent this problem:
$${\text{Minimize: }} f_{1} \left( x \right) = 4.9 \times \left( {10} \right)^{{\left( { - 5} \right)}} \times \left( {x\left( 2 \right)^{\left( 2 \right)} - x\left( 1 \right)^{\left( 2 \right)} } \right) \times \left( {x\left( 4 \right) - 1} \right)$$
(B.13)
$${\text{Minimize: }} f_{2} \left( x \right) = \left( {9.82 \times \left( {10} \right)^{\left( 6 \right)} } \right)) \times \left( {x\left( 2 \right))^{\left( 2 \right)} - x\left( 1 \right)^{\left( 2 \right)} } \right))/\left( {\left( {x\left( 2 \right))^{\left( 3 \right)} - x\left( 1 \right)^{\left( 3 \right)} } \right) \times x\left( 4 \right) \times x\left( 3 \right)} \right)$$
(B.14)
$$g_{1} \left( x \right) = 20 + x\left( 1 \right) - x\left( 2 \right)$$
(B.15)
$$g_{2} \left( x \right) = 2.5 + \left( {x\left( 4 \right) + 1} \right) - 30$$
(B.16)
$$g_{3} \left( x \right) = \left( {x\left( 3 \right)} \right)/\left( {3.14 \times \left( {x\left( 2 \right)^{2} - x\left( 1 \right)^{2} } \right)^{2} } \right) - 0.4$$
(B.17)
$$g_{4} \left( x \right) = \left( {2.22 \times \left( {10} \right)^{{\left( { - 3} \right)}} \times x\left( 3 \right) \times \left( {x\left( 2 \right)^{3} - x\left( 1 \right)^{3} } \right)} \right)/\left( {\left( {x\left( 2 \right)^{2} - x\left( 1 \right)^{2} } \right)^{2} } \right) - 1$$
(B.18)
$$g_{5} \left( x \right) = 900 - \left( {2.66 \times \left( {10} \right)^{{\left( { - 2} \right)}} \times x\left( 3 \right) \times x\left( 4 \right) \times \left( {x\left( 2 \right)^{3} - x\left( 1 \right)^{3} } \right)} \right)/\left( {\left( {x\left( 2 \right)^{2} - x\left( 1 \right)^{2} } \right)^{2} } \right)$$
(B.19)
$$55 \le x_{1} \le 80, 75 \le x_{2} \le 110$$
$$1000 \le x_{3} \le 3000, 2 \le x_{4} \le 20$$
Speed reducer design problem
This issue [51, 53] contains two objectives such as weight (\({f}_{1}\)) and stress (\({f}_{2}\)), which are to be minimized. The problem may represent with a diagram as given in Fig. 10. Also, the problem has eleven constraints with seven numbers of design variables such as width of gear face (\({x}_{1}\)), teeth module (\({x}_{2}\)), pinion teeth number (\({x}_{3}\) numeral variable), a distance between of bearings 1 (\({x}_{4}\)), a distance of bearings 2 (\({x}_{5}\)), shaft 1 diameter (\({x}_{6}\)), and shaft 2 diameter (\({x}_{7}\)). The equations may clearly represent this problem as below:
$${\text{Minimize: }} f_{1} \left( x \right) = 0.7854 \times x\left( 1 \right) \times x(2)^{2} \times (3.3333 \times x\left( {3)^{2} + 14.9334 \times x\left( 3 \right)} \right) \ldots$$
$$- 43.0934) - 1.508 \times x\left( 1 \right) \times (x(6)^{2} + x(7)^{2}$$
(B.20)
$${\text{Minimize}}:{ }f_{2} \left( x \right) = (({\text{sqrt}}\left( {\left( {\left( {745*x\left( 4 \right)} \right)/x\left( 2 \right)*x\left( 3 \right)} \right))^{2} + 19.9e6} \right)/\left( {0.1*x\left( {6)^{3} } \right)} \right)$$
(B.21)
$${\text{where }} g_{1} \left( x \right) = 27/(x\left( 1 \right) \times x\left( {2)^{2} \times x\left( 3 \right)} \right) - 1$$
(B.22)
$$g_{2} \left( x \right) = 397.5/(x\left( 1 \right) \times x(2)^{2} \times x\left( {3)^{2} } \right) - 1$$
(B.23)
$$g_{3} \left( x \right) = (1.93 \times (x\left( {4)^{3} } \right)/(x\left( 2 \right) \times x\left( 3 \right) \times x\left( {6)^{4} } \right) - 1$$
(B.24)
$$g_{4} \left( x \right) = (1.93 \times (x\left( {5)^{3} } \right)/(x\left( 2 \right) \times x\left( 3 \right) \times x\left( {7)^{4} } \right) - 1$$
(B.25)
$$g_{5} \left( x \right) = (({\text{sqrt}}\left( {\left( {745 \times x\left( 4 \right)} \right)/x\left( 2 \right) \times x\left( {3)))^{2} + 16.9e6} \right)} \right)/\left( {110 \times x\left( {6)^{3} } \right)} \right) - 1$$
(B.26)
$$g_{6} \left( x \right) = (({\text{sqrt}}\left( {\left( {745 \times x\left( 4 \right)} \right)/x\left( 2 \right) \times x\left( {3)))^{2} + 157.5e6} \right)} \right)/\left( {85 \times x\left( {7)^{3} } \right)} \right) - 1$$
(B.27)
$$g_{7} \left( x \right) = \left( {\left( {x\left( 2 \right) \times x\left( 3 \right)} \right)/40} \right) {-} 1$$
(B.28)
$$\tau = {\text{sqrt}}\left( {\alpha^{2} + 2 \times \alpha \times \beta \times \frac{x\left( 2 \right)}{{2 \times D}} + \beta^{2} } \right)$$
(B.29)
$$\sigma = \frac{504000}{{x\left( 4 \right) \times x(3)^{2} }}$$
(B.30)
$${\text{tmpf}} = 4.013 \times \frac{{30 \times 10^{6} }}{196}$$
(B.31)
$$P = {\text{tmpf}} \times {\text{sqrt}}\left( {x(3)^{2} \times \frac{{x(4)^{6} }}{36}} \right) \times \left( {1 - x\left( 3 \right) \times \frac{{{\text{sqrt}}\left( {\frac{30}{{48}}} \right)}}{28}} \right)$$
(B.32)