A Comparative Study of Metaheuristic Algorithms for Reliability-Based Design Optimization Problems

Abstract

The ever-increasing demands for resource-saving, engineering technology progress, and environmental protection stimulate the progress of the progressive design method. As an excellent promising design method for dealing with the inevitable uncertainty factors, reliability-based design optimization (RBDO) is capable of offering reliable and robust results and minimizing the cost under the prescribed uncertainty level, which can provide a trade-off between economy and safety. However, the primary challenges, including global convergence capacity and complicated mixed design variable type, hinder the wider application of RBDO. This study presents a comprehensive work on the application of ten popular and recent metaheuristic algorithms of five engineering problems. Furthermore, we focus on the RBDO equip with metaheuristic algorithms about its global convergence, robustness, accuracy, and computational speed. This paper also presents the major difference of convergence property between metaheuristic algorithms and gradient algorithms. The detailed statement of this study presents the state-of-the-art in RBDO to demonstrate its crucial technologies and great challenges, as well as the beneficial future development direction.

Appendix

1.1 RBDO for Automobile Side Impact Case

The RBDO model for automobile side impact case is expressed as:

$$\begin{aligned} & {\text{find}}\quad {\mathbf{d}} = [d_{1} ,d_{2} ,d_{3} ,d_{4} ,d_{5} ,d_{6} ,d_{7} ,d_{8} ,d_{9} ]^{\text{T}} \\ & \hbox{min} \quad {\text{obj}}({\varvec{\upmu}}_{{\mathbf{x}}} ) \\ & {\text{s.t.}} \quad \quad P(F_{AL} \le 1 \,{\mathbf{KN}}) \ge R_{1} \\ & \quad \quad \quad P(D_{up} \le 32\,{\text{cm}}) \ge R_{2} \\ & \quad \quad \quad P(D_{mid} \le 32\,{\text{cm}}) \ge R_{3} \\ & \quad \quad \quad P(D_{low} \le 32\,{\text{cm}}) \ge R_{4} \\ & \quad \quad \quad P({\text{VC}}_{up} \le 0.32\,{\text{cm}}) \ge R_{5} \\ & \quad \quad \quad P({\text{VC}}_{mid} \le 0.32\,{\text{cm}}) \ge R_{6} \\ & \quad \quad \quad P({\text{VC}}_{low} \le 0.32\,{\text{cm}}) \ge R_{7} \\ & \quad \quad \quad P(F_{\text{ps}} \le 4.0 {\mathbf{KN}}) \ge R_{8} \\ & \quad \quad \quad P(V_{\text{B - Pillar}} \le 9.9\,{\text{m/s}}) \ge R_{9} \\ & \quad \quad \quad P(V_{\text{door}} \le 15.69\,{\text{m/s}}) \ge R_{10} \\ & \quad \quad \quad {\varvec{\upmu}}_{i}^{L} \le {\varvec{\upmu}}_{i} \le {\varvec{\upmu}}_{i}^{U} ,\quad i = 1\sim 7 \\ \end{aligned}$$

where

$$\begin{aligned} & {\text{obj}} = 1.98 + 4.9x_{1} + 6.67x_{2} + 6.98x_{3} + 4.01x_{4} + 1.78x_{5} + 2.73x_{7} \\ & F_{AL} = 1.16 - 0.3717x_{2} x_{4} - 0.00931x_{2} x_{10} - 0.484x_{3} x_{9} + 0.01343x_{6} x_{10} \\ & D_{up} = 28.98 + 3.818x_{3} - 4.2x_{1} x_{2} + 0.0207x_{5} x_{10} + 6.63x_{6} x_{9} - 7.77x_{7} x_{8} + 0.32x_{9} x_{10} \\ & D_{mid} = 33.86 + 2.95x_{3} + 0.1792x_{10} - 5.057x_{1} x_{2} - 11x_{2} x_{8} - 0.0215x_{5} x_{10} - 9.98x_{7} x_{8} + 22x_{8} x_{9} \\ & D_{low} = 46.36 - 9.9x_{2} - 12.9x_{1} x_{8} + 0.1107x_{3} x \\ & VC_{up} = 0.261 - 0.0159x_{1} x_{2} - 0.188x_{1} x_{8} - 0.019x_{2} x_{7} + 0.0144x_{3} x_{5} + 0.8757x_{5} x_{10} + 0.08045x_{6} x_{9} \\ & \quad \quad \quad + 0.00139x_{8} x_{11} + 1.575(10^{ - 6} )x_{10} x_{11} \\ & VC_{mid} = 0.214 + 0.00817x_{5} - 0.131x_{1} x_{8} - 0.0704x_{1} x_{9} + 0.03099x_{2} x_{6} - 0.018x_{2} x_{7} \\ & \quad \quad \quad + 0.0208x_{3} x_{8} + 0.121x_{3} x_{9} - 0.00364x_{5} x_{6} + 0.0007715x_{5} x_{10} - 0.0005354x_{6} x_{10} \\ & \quad \quad \quad + 0.00121x_{8} x_{11} + 0.00184x_{9} x_{10} - 0.02x_{2} x_{2} \\ & VC_{low} = 0.74 - 0.61x_{2} - 0.163x_{3} x_{8} + 0.001232x_{3} x_{10} - 0.166x_{7} x_{9} + 0.227x_{2} x_{2} \\ & F_{ps} = 4.72 - 0.5x_{4} - 0.19x_{2} x_{3} - 0.0122x_{4} x_{10} + 0.009325x_{6} x_{10} + 0.000191x_{11} x_{11} \\ & V_{B - Pillar} = 10.58 - 0.674x_{1} x_{2} - 1.95x_{2} x_{8} + 0.02054x_{3} x_{10} - 0.0198x_{4} x_{10} + 0.028x_{6} x_{10} \\ & V_{door} = 16.45 - 0.489x_{3} x_{7} - 0.843x_{5} x_{6} + 0.0432x_{9} x_{10} - 0.0556x_{9} x_{11} - 0.000786x_{11} x_{11} \\ \end{aligned}$$

where $R_{1} { = }R_{2} = \cdots = R_{11} = \varPhi (3)$

$$x_{i} \sim N(1,0.03^{2} ) ,i = 1 - 7, i \ne 5;\,x_{5} \sim N(2,0.05^{2} );\,x_{i} \sim N(0.192,0.06^{2} ),i = 8\,{\text{and}}\,9;$$

$$x_{i} \sim N(0,10^{2} ) , i = 10 {\text{and}} 11$$

$$0.5 \le x_{i} \le 1.5, i = 1,3,4; 0.45 \le x_{2} \le 1.35; 0.875 \le x_{5} \le 2.625; 0.4 \le x_{6} \le 1.2; 0.4 \le x_{7} \le 1.2\,x_{8} ,x_{9} \in (0.192,0.345).$$

1.2 RBDO for a Bolted Rim

The RBDO model for bolted rim is formulated as (Table 8):

Table 8 Discrete values of bolts d

$$\begin{aligned} & {\text{find}}\quad {\mathbf{d}} = [R_{B} ,M,d,N]^{\text{T}} \\ & \hbox{min} \quad f\left( {\mathbf{d}} \right) = \beta_{1} \left( {\frac{N}{{N_{m} }}} \right) + \beta_{2} \left( {\frac{{R_{B} + \phi_{4} (d) + c}}{{R_{m} }}} \right) + \beta_{3} \left( {\frac{M}{{M_{T} }}} \right) \\ & {\text{s.t.}}\quad P[G_{i} ({\mathbf{x}}) > 0] \ge \varPhi (\beta_{i}^{t} );\quad i = 1 - 11. \\ & G_{1} ({\mathbf{x}}) = 1 - \frac{\alpha M}{{NR_{B} K(d)}} \ge 0,\quad \quad G_{2} ({\mathbf{x}}) = \frac{{2\pi R_{B} }}{{\phi_{5} (d)N}} - 1 \ge 0 \\ & G_{3} ({\mathbf{x}}) = \frac{{R_{B} }}{{\phi_{4} (d)}} + R_{M} - 1 \ge 0,\quad \quad G_{4} ({\mathbf{x}}) = N_{\hbox{max} } - N \ge 0 \\ & G_{5} ({\mathbf{x}}) = R_{\hbox{max} } - R_{B} \ge 0,\quad \quad G_{6} ({\mathbf{x}}) = N - N_{M} \ge 0 \\ & G_{7} ({\mathbf{x}}) = R_{B} - R_{M} \ge 0,\quad \quad G_{8} ({\mathbf{x}}) = M_{\hbox{max} } - M \ge 0 \\ & G_{9} ({\mathbf{x}}) = M - M_{T} \ge 0,\quad \quad G_{10} ({\mathbf{x}}) = 24 - d \ge 0 \\ & G_{11} ({\mathbf{x}}) = d - 6 \ge 0 \\ & \beta_{1}^{t} = \beta_{2}^{t} = \cdot \cdot \cdot = \beta_{11}^{t} = 3.0,\; \\ \end{aligned}$$

where $R_{B} \sim N(60,1),M\sim N(60,1)$

$$K(d) = \frac{{0.9f_{m} R_{e} \pi \left( {\phi_{1} (d)} \right)^{2} }}{{4\sqrt {1 + 3(0.16\phi_{3} (d)f_{1} /\phi_{1} (d))^{2} } }}, \, M_{T} = 40\,{\text{Nm}}, \, M_{\hbox{max} } = 1000\,{\text{Nm}},f_{m} = 0.15,$$

$$\, f_{1} = 0.15,\alpha = 1.5, \, R_{e} = 627\,{\text{MPa}}, \, N_{M} = 8, \, N_{\hbox{max} } = 100, \, R_{M} = 50\,{\text{mm}}, \, R_{\hbox{max} } = 1000\,{\text{mm}},c = 5\,{\text{mm}}, \, \beta_{1} = \beta_{2} = \beta_{3} = 1$$

$$6 \le d \le 24 , { }8 \le N \le 100, \, 50 \le R_{B} \le 100,40 \le M \le 100.$$

1.3 RBDO for Spur Speed Reducer

The RBDO model for spur speed reducer is depicted as:

$$\begin{aligned} & {\text{find}}\quad {\mathbf{d}} = [\mu_{{x_{1} }} ,\mu_{{x_{2} }} ,\mu_{{x_{3} }} ,\mu_{{x_{4} }} ,\mu_{{x_{5} }} ,\mu_{{x_{6} }} ,\mu_{{x_{7} }} ]^{\text{T}} \\ & \hbox{min} \quad f({\mathbf{d}}) = 0.7854\mu_{{x_{1} }} \mu_{{x_{2} }}^{2} \left( {3.3333\mu_{{x_{3} }}^{2} + 14.9334\mu_{{x_{3} }}^{{}} - 43.0934} \right) \\ & \quad \quad \; - 1.508\mu_{{x_{1} }} \left( {\mu_{{x_{6} }}^{2} + \mu_{{x_{7} }}^{2} } \right) + 7.477\left( {\mu_{{x_{6} }}^{3} + \mu_{{x_{7} }}^{3} } \right) + 0.7854\left( {\mu_{{x_{4} }} \mu_{{x_{6} }}^{2} + \mu_{{x_{5} }} \mu_{{x_{7} }}^{2} } \right) \\ & {\text{s.t.}} \quad \quad {\text{P}}\left[ {G_{i} ({\mathbf{x}}) > 0} \right] \ge \varPhi \left( {\beta_{i}^{t} } \right); i = 1 - 11. \\ & G_{1} ({\mathbf{x}}) = \frac{27}{{x_{1} x_{2}^{2} x_{3} }} - 1, G_{2} ({\mathbf{x}}) = \frac{397.5}{{x_{1} x_{2}^{2} x_{3}^{2} }} - 1 \\ & G_{3} ({\mathbf{x}}) = \frac{{1.93x_{4}^{3} }}{{x_{2} x_{3} x_{6}^{4} }} - 1, G_{4} ({\mathbf{x}}) = \frac{{1.93x_{5}^{3} }}{{x_{2} x_{3} x_{7}^{4} }} - 1 \\ & G_{5} ({\mathbf{x}}) = \frac{{\sqrt {\left( {{{745x_{4} } \mathord{\left/ {\vphantom {{745x_{4} } {(x_{2} x_{3} )}}} \right. \kern-0pt} {(x_{2} x_{3} )}}} \right)^{2} + 16.9 \times 10^{6} } }}{{0.1x_{6}^{3} }} - 1100 \\ & G_{6} ({\mathbf{x}}) = \frac{{\sqrt {\left( {{{745x_{4} } \mathord{\left/ {\vphantom {{745x_{4} } {(x_{2} x_{3} )}}} \right. \kern-0pt} {(x_{2} x_{3} )}}} \right)^{2} + 157.5 \times 10^{6} } }}{{0.1x_{7}^{3} }} - 850 \\ & G_{7} ({\mathbf{x}}) = x_{2} x_{3} - 40, G_{8} ({\mathbf{x}}) = 5 - \frac{{x_{1} }}{{x_{2} }} \\ & G_{9} ({\mathbf{x}}) = \frac{{x_{1} }}{{x_{2} }} - 12, G_{10} ({\mathbf{x}}) = \frac{{1.5x_{6} + 1.9}}{{x_{4} }} - 1 \\ & G_{11} ({\mathbf{x}}) = \frac{{1.1x_{7} + 1.9}}{{x_{5} }} - 1 \\ & \beta_{1}^{t} = \beta_{2}^{t} = \cdots = \beta_{11}^{t} = 3.0 \\ & 2.6 \le \mu_{{x_{1} }} \le 3.6,\;0.7 \le \mu_{{x_{2} }} \le 0.8,17 \le \mu_{{x_{3} }} \le 28 \\ & 7.3 \le \mu_{{x_{4} }} \le 8.3,\;\;7.3 \le \mu_{{x_{5} }} \le 8.3 \\ & 2.9 \le \mu_{{x_{6} }} \le 3.9,\;5.0 \le \mu_{{x_{7} }} \le 5.5 \\ & x_{i} \sim N(\mu_{{x_{i} }} , 0.005^{2} ),\quad {\text{for}} \;i = 1 - 7. \\ \end{aligned}$$

1.4 RBDO for Welded Beam

The RBDO model for welded beam is expressed as:

$$\begin{aligned} & {\text{find}}\quad {\mathbf{d}} = [d_{1} ,d_{2} ,d_{3} ,d_{4} ] \\ & \hbox{min} \quad \,f\left( {{\mathbf{d}},{\mathbf{z}}} \right) = c_{1} d_{1}^{2} d_{2} + c_{2} d_{3} d_{4} (z_{2} + d_{2} ) \\ & {\text{s.t.}}\quad \; P(G_{i} \left( {{\mathbf{x}},{\mathbf{z}}} \right) > 0) \ge \varPhi \left( {\beta_{i}^{t} } \right),\,\,i = 1, \ldots ,5 \\ & {\text{where}}\,\,G_{1} \left( {{\mathbf{x}},{\mathbf{z}}} \right) = {{\tau ({\mathbf{x}},{\mathbf{z}})} \mathord{\left/ {\vphantom {{\tau ({\mathbf{x}},{\mathbf{z}})} {z_{6} }}} \right. \kern-0pt} {z_{6} }} - 1,\;\;G_{2} \left( {{\mathbf{x}},{\mathbf{z}}} \right) = {{\sigma \left( {{\mathbf{x}},{\mathbf{z}}} \right)} \mathord{\left/ {\vphantom {{\sigma \left( {{\mathbf{x}},{\mathbf{z}}} \right)} {z_{7} }}} \right. \kern-0pt} {z_{7} }} - 1, \\ & \quad \quad G_{3} \left( {{\mathbf{x}},{\mathbf{z}}} \right) = {{x_{1} } \mathord{\left/ {\vphantom {{x_{1} } {x_{4} }}} \right. \kern-0pt} {x_{4} }} - 1,\;\;\;G_{4} \left( {{\mathbf{x}},{\mathbf{z}}} \right) = {{\delta \left( {{\mathbf{x}},{\mathbf{z}}} \right)} \mathord{\left/ {\vphantom {{\delta \left( {{\mathbf{x}},{\mathbf{z}}} \right)} {z_{5} - 1,}}} \right. \kern-0pt} {z_{5} - 1,}} \\ & \quad \quad G_{5} \left( {{\mathbf{x}},{\mathbf{z}}} \right) = 1 - P_{c} \left( {{\mathbf{x}},{\mathbf{z}}} \right)/z_{1} , \\ & \quad \quad \tau \left( {{\mathbf{x}},{\mathbf{z}}} \right) = \{ t\left( {{\mathbf{x}},{\mathbf{z}}} \right)^{2} + \frac{{2t\left( {{\mathbf{x}},{\mathbf{z}}} \right)tt\left( {{\mathbf{x}},{\mathbf{z}}} \right)X_{2} }}{{2R\left( {\mathbf{x}} \right)}} + tt\left( {{\mathbf{x}},{\mathbf{z}}} \right)^{2} \}^{1/2} , \\ & \quad \quad t\left( {{\mathbf{x}},{\mathbf{z}}} \right) = \frac{{z_{1} }}{{\sqrt 2 x_{1} x_{2} }},\;\;tt\left( {{\mathbf{x}},{\mathbf{z}}} \right) = \frac{{M\left( {{\mathbf{x}},{\mathbf{z}}} \right)R\left( {\mathbf{x}} \right)}}{{J\left( {\mathbf{x}} \right)}}, \\ & \quad \quad \sigma \left( {{\mathbf{x}},{\mathbf{z}}} \right) = \frac{{6z_{1} z_{2} }}{{x_{3}^{2} x_{4} }},\;\;\delta \left( {{\mathbf{x}},{\mathbf{z}}} \right) = \frac{{4z_{1} z_{2}^{3} }}{{z_{3} x_{3}^{3} x_{4} }}, \\ & \quad \quad M\left( {{\mathbf{x}},{\mathbf{z}}} \right) = z_{1} \left( {z_{2} + \frac{{x_{2} }}{2}} \right),\;\;R\left( {\mathbf{x}} \right) = \left( {\frac{{\sqrt {x_{2}^{2} + (x_{1} + x_{3} )^{2} } }}{2}} \right), \\ & \quad \quad J\left( {\mathbf{x}} \right) = \sqrt 2 x_{1} x_{2} \left\{ {{{x_{2}^{2} } \mathord{\left/ {\vphantom {{x_{2}^{2} } {12}}} \right. \kern-0pt} {12}} + {{\left( {x_{1} + x_{3} } \right)^{2} } \mathord{\left/ {\vphantom {{\left( {x_{1} + x_{3} } \right)^{2} } 4}} \right. \kern-0pt} 4}} \right\}, \\ & \quad \quad P_{c} \left( {{\mathbf{x}},{\mathbf{z}}} \right) = \frac{{4.013x_{3} x_{4}^{3} \sqrt {z_{3} z_{4} } }}{{6z_{2}^{2} }}\left( {1 - \frac{{x_{3} }}{{4z_{2} }}\sqrt {\frac{{z_{3} }}{{z_{4} }}} } \right), \\ & \quad \quad \beta_{1}^{t} = \beta_{2}^{t} = \cdots = \beta_{5}^{t} = 3.0,\; \\ & \quad \quad 3.175 \le d_{1} \le 50.8,\;0 \le d_{2} \le 254, \\ & \quad \quad 0 \le d_{3} \le 254,\;\;0 \le d_{4} \le 50.8 \\ & \quad \quad x_{i} \sim N(d_{i} ,0.1693^{2} )\quad {\rm for}\;i = 1,2, \\ & \quad \quad x_{i} \sim N(d_{i} ,0.0107^{2} )\quad {\rm for}\;i = 3,4. \\ \end{aligned}$$

1.5 RBDO for Stiffened Shell

The RBDO for stiffened shell is expressed as:

$$\begin{aligned} & {\text{find}} \quad {\mathbf{d}} = [t_{s}^{C} ,\,t_{r}^{C} ,h_{{}}^{C} ,N_{c} ,N_{a} ]^{\text{T}} \\ & \mathop {\hbox{min} }\limits_{{}} \quad f({\mathbf{d}}) \\ & {\text{s.t.}}\quad P({\text{load(}}t_{s} ,t_{r} ,h,N_{c} ,N_{a} ,E,\nu ) - 1.2747 \times 10^{7} > 0) \ge \varPhi (\beta_{{}}^{t} ) \\ & \quad \quad 2.5 \le t_{s} \le 5.5,6.0 \le t_{r} \le 12.0,9.0 \le h \le 23.0 \\ & \quad \quad 10 \le N_{c} \le 38;80 \le N_{a} \le 130,\beta_{{}}^{t} = 3. \\ \end{aligned}$$