# Constrained Mixed-Variable Design Optimization Based on Particle Swarm Optimizer with a Diversity Classifier for Cyclically Neighboring Subpopulations

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

- (I)
- Most engineering design tasks include multiple real-life physical constraints and thus necessitate PSO techniques that can account for these. Furthermore, such physical constraints are, in general, treated as hard constraints that should be satisfied by any feasible solution found via the optimization procedure.
- (II)
- A large fraction of design problems in engineering fields belong in the category of mixed-integer-discrete-continuous (MIDC) optimization problems and thus particular care has to be taken to find feasible solutions, which makes design tasks more challenging.

## 2. Engineering Design Problem and Particle Swarm Optimizer

## 3. PSO with a Diversity Classifier for Cyclically Neighboring Subpopulations

- (I)
- Initialize the entire swarm. Then, the particles evolve in the $\mathbb{D}$-dimensional search space, regardless of the variable types in ${\mathit{x}}_{i}\phantom{\rule{3.33333pt}{0ex}}(:={({\mathit{x}}^{\mathrm{I}},\phantom{\rule{3.33333pt}{0ex}}{\mathit{x}}^{\mathrm{D}},\phantom{\rule{3.33333pt}{0ex}}{\mathit{x}}^{\mathrm{C}})}^{T})$, of (5), via (2) and (10) with ${\mathit{x}}_{\mathrm{sbest},i}^{k}$. As a result, all elements (i.e., all design variables) in ${\mathit{x}}_{i}$ take floating-point values during the current stage.
- (II)
- For $j=1,2,\cdots ,{n}_{\mathrm{I}}$, the variable ${x}_{i,j}$, which is the jth entry of ${\mathit{x}}_{i}\in \mathbb{R}$, denotes the integer design variable ${x}_{i,j}^{\mathrm{I}}$ and thus must take an integer value. Let $\mathrm{INT}\left({x}_{i,j}\right)$ denote the nearest integer of ${x}_{i,j}\in \mathbb{R}$. i.e., ${x}_{i,j}$ is rounded to its nearest integer. It then follows that ${x}_{i,j}^{\mathrm{I}}\leftarrow \mathrm{INT}\left({x}_{i,j}\right)$ is performed. Similarly, the discrete design variable ${x}_{i,{n}_{\mathrm{I}}+\ell}^{\mathrm{D}}$ for $\ell =1,2,\cdots ,{n}_{\mathrm{D}}$ takes the value of $\mathrm{DIS}\left({x}_{i,{n}_{\mathrm{I}}+\ell}\right)$ as ${x}_{i,{n}_{\mathrm{I}}+\ell}^{\mathrm{D}}\leftarrow \mathrm{DIS}\left({x}_{i,{n}_{\mathrm{I}}+\ell}\right)$. Here, $\mathrm{DIS}\left({x}_{i,{n}_{\mathrm{I}}+\ell}\right)$ indicates the nearest discrete value that ${x}_{i,{n}_{\mathrm{I}}+\ell}$ takes in the given data set of discrete design values.
- (III)
- Fitness of individual particle with positional vector $({\mathit{x}}_{i}^{\mathrm{I}},{\mathit{x}}_{i}^{\mathrm{D}},{\mathit{x}}_{i}^{\mathrm{C}})$, is evaluated based on the objective function $\mathcal{L}\left(\mathit{x}\right)$, as in (8) subject to no constraint functions. Finally, ${\mathit{x}}_{\mathrm{pbest},i}$ and ${\mathit{x}}_{\mathrm{sbest},i}^{k}$ in (9), are calculated.

## 4. Numerical Experimentation

#### 4.1. Optimal Design of Pressure Vessel

#### 4.2. Optimal Design of Reinforced Concrete Beam

#### 4.3. Optimal Design of Helical Compression Spring

#### 4.4. Optimal Design of Belleville Spring

#### 4.5. Optimal Design of Speed Reducer

#### 4.6. Optimal Design of Stepped Cantilever Beam

#### 4.7. Optimal Design of Rolling Element Bearing

#### 4.8. Car Side Impact Design Problem

#### 4.9. Discussions

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of the proposed diversity-guided particle swarm optimizer (PSO) with a cyclic network mechanism.

**Figure 2.**Flowchart of the PSO scheme with a diversity classifier for cyclically neighboring subpopulations for solving the constrained MIDC optimization problems.

**Figure 10.**Finite element model utilized in the car side impact problem [1].

**Figure 11.**Distributions of $f\left({\mathit{x}}^{*}\right)$ corresponding to ${n}_{s}/{n}_{p}$ for eight numerical examples.

Method | [32] PSO-GA | [31] HS | [33] SA-DS | [1] FA | [2] PSO | Present Study PSO (${\mathit{n}}_{\mathit{s}}\mathbf{=}\mathbf{16}$) |
---|---|---|---|---|---|---|

${x}_{1}$ (l) | 221.365487 | 221.36553 | 207.22555 | 221.36547 | 221.365548 | 221.3654714 |

${x}_{2}$ (r) | 38.860102 | 38.86010 | 39.80962 | 38.86010 | 38.860099 | 38.8601036 |

${x}_{3}$ (${t}_{s}$) | 0.7500 | 0.75 | 0.7683 | 0.75 | 0.75000 | 0.7500000 |

${x}_{4}$ (${t}_{h}$) | 0.3750 | 0.375 | 0.3797 | 0.375 | 0.37500 | 0.3750000 |

${h}_{1}\left({\mathit{x}}^{*}\right)$ | −0.0000 | −0.0000 | −0.0000 | −0.0000 | −0.000000 | −0.0000000 |

${h}_{2}\left({\mathit{x}}^{*}\right)$ | −0.0043 | −0.0043 | −0.0000 | −0.0043 | −0.004275 | −0.0042746 |

${h}_{3}\left({\mathit{x}}^{*}\right)$ | 0.0446 | 0.2713 | −10.7065 | −0.0134 | −0.000190 | −0.0255540 |

${h}_{4}\left({\mathit{x}}^{*}\right)$ | −18.6345 | −18.6345 | −32.7744 | −18.6345 | −18.634452 | −18.6345290 |

Best objective value | 5850.383064 | 5849.76169 | 5868.76484 | 5850.38306 | 5850.38376 | 5850.38306 |

Objective deviation ^{a} | $1.0+6.837\times {10}^{-10}$ | $1.0-1.062\times {10}^{-4}$ | $1.0+3.142\times {10}^{-3}$ | 1.0 | $1.0+1.197\times {10}^{-7}$ | 1.0 |

Feasibility | Infeasible | Infeasible | Feasible | Feasible | Feasible | Feasible |

Worst objective value | N/A ^{b} | N/A | 6804.328100 | 6258.96825 | 5850.591797 | 5850.38308 |

Average objective value | N/A | N/A | 6164. 585867 | 5937.33790 | N/A | 5850.38306 |

Standard deviation | N/A | N/A | 257.473670 | 164.54747 | N/A | $3.7\times {10}^{-6}$ |

Function evaluations | 100,000 | 200,000 | N/A | 25,000 | 31,436–124,968 | 50,000 |

^{a}The ratio of the best objective value of each method to the lowest one among all methods.

^{b}N/A denotes non-available.

Bar Type | ${\mathit{A}}_{\mathit{s}}$ (${\mathbf{in}}^{2}$) | Bar Type | ${\mathit{A}}_{\mathit{s}}$ (${\mathbf{in}}^{2}$) | Bar | ${\mathit{A}}_{\mathit{s}}$ (${\mathbf{in}}^{2}$) | Bar Type | ${\mathit{A}}_{\mathit{s}}$ (${\mathbf{in}}^{2}$) |
---|---|---|---|---|---|---|---|

1#4 | 0.2 | 6#5 | 1.86 | 9#6 | 3.96 | 12#7 | 7.2 |

1#5 | 0.31 | 10#4, 2#9 | 2 | 4#9 | 4 | 13#7 | 7.8 |

2#4 | 0.4 | 7#5 | 2.17 | 13#5 | 4.03 | 10#8 | 7.9 |

1#6 | 0.44 | 11#4, 5#6 | 2.2 | 7#7 | 4.2 | 8#9 | 8 |

3#4, 1#7 | 0.6 | 3#8 | 2.37 | 14#5 | 4.34 | 14#7 | 8.4 |

2#5 | 0.62 | 12#4, 4#7 | 2.4 | 10#6 | 4.4 | 11#8 | 8.69 |

1#8 | 0.79 | 8#5 | 2.48 | 15#5 | 4.65 | 15#7 | 9 |

4#4 | 0.8 | 13#4 | 2.6 | 6#8 | 4.74 | 12#8 | 9.48 |

2#6 | 0.88 | 6#6 | 2.64 | 8#7 | 4.8 | 13#8 | 10.27 |

3#5 | 0.93 | 9#5 | 2.79 | 11#6 | 4.84 | 11#9 | 11 |

5#4, 1#9 | 1 | 14#4 | 2.8 | 5#9 | 5 | 14#8 | 11.06 |

6#4, 2#7 | 1.2 | 15#4, 5#7, 3#9 | 3 | 12#6 | 5.28 | 15#8 | 11.85 |

4#5 | 1.24 | 7#6 | 3.08 | 9#7 | 5.4 | 12#9 | 12 |

3#6 | 1.32 | 10#5 | 3.10 | 7#8 | 5.53 | 13#9 | 13 |

7#4 | 1.4 | 4#8 | 3.16 | 13#8 | 5.72 | 14#9 | 14 |

5#5 | 1.55 | 11#5 | 3.41 | 10 7, 6#9 | 6 | 15#9 | 15 |

2#8 | 1.58 | 8#6 | 3.52 | 14#6 | 6.16 | ||

8#4 | 1.6 | 6#7 | 3.6 | 8#8 | 6.32 | ||

4#6 | 1.76 | 12#5 | 3.72 | 15#6, 11#7, 7#9 | 6.6 | ||

9#4, 3#7 | 1.8 | 5#8 | 3.95 | 9#8 | 7.11 |

Reference Method | [34] SD-RC ^{a} | [35] | [36] BFO | [37] | [1] FA | Present Study PSO (${\mathit{n}}_{\mathit{s}}=8$) | ||
---|---|---|---|---|---|---|---|---|

GHN-ALM ^{b} | GHN-EP ^{c} | GA | GA-FL | |||||

${x}_{1}$ (${A}_{s}$) | 7.8 | 6.6 | 6.32 | N/A | 7.20 | 6.16 | 6.32 | 6.32 |

${x}_{2}$ (b) | 31 | 33 | 34 | N/A | 32 | 35 | 34 | 34 |

${x}_{3}$ (h) | 7.79 | 8.495227 | 8.637180 | N/A | 8.0451 | 8.7500 | 8.5000 | 8.5000 |

${h}_{1}\left({\mathit{x}}^{*}\right)$ | −0.0205 | −0.1155 | −0.0635 | N/A | −0.0224 | 0 | 0 | 0 |

${h}_{2}\left({\mathit{x}}^{*}\right)$ | −4.2012 | 0.0159 | −0.7745 | N/A | −2.8779 | −3.6173 | −0.2241 | −0.22409 |

Best objective value | 374.2 | 362.2455 | 362.00648 | 376.2977 | 366.1459 | 364.8541 | 359.2080 | 359.2080 |

Objective deviation | $1.0+4.174\times {10}^{-2}$ | $1.0+8.456\times {10}^{-3}$ | $1.0+7.791\times {10}^{-3}$ | $1.0+4.758\times {10}^{-2}$ | $1.0+1.931\times {10}^{-2}$ | $1.0+1.572\times {10}^{-2}$ | 1.0 | 1.0 |

Feasibility | Feasible | Infeasible | Feasible | N/A | Feasible | Feasible | Feasible | Feasible |

Worst objective value | N/A | N/A | N/A | N/A | N/A | N/A | 669.150 | 359.2080 |

Average objective value | N/A | N/A | N/A | N/A | 371.5417 | 365.8046 | 460.706 | 359.2080 |

Standard deviation | N/A | N/A | N/A | N/A | N/A | N/A | 80.73870 | 0.0000 |

Function evaluations | 396 | N/A | N/A | 100,000 | 100,000 | 30,000 | 25,000 | 20,000 |

^{a}Hybrid discrete steepest descent and rotating coordinate directions methods.

^{b}Generalized Hopfield network-based augmented Lagrange multiplier approach.

^{c}GHN-based extended penalty approach.

Allowable Wire Diameter (in) | ||||||
---|---|---|---|---|---|---|

0.0090 | 0.0095 | 0.0104 | 0.0118 | 0.0128 | 0.0132 | 0.0140 |

0.0150 | 0.0162 | 0.0173 | 0.0180 | 0.0200 | 0.0230 | 0.0250 |

0.0280 | 0.0320 | 0.0350 | 0.0410 | 0.0470 | 0.0540 | 0.0630 |

0.0720 | 0.0800 | 0.0920 | 0.1050 | 0.1200 | 0.1350 | 0.1480 |

0.1620 | 0.1770 | 0.1920 | 0.2070 | 0.2250 | 0.2440 | 0.2630 |

0.2830 | 0.3070 | 0.3310 | 0.3620 | 0.3940 | 0.4375 | 0.5000 |

Notation | Description | Value |
---|---|---|

${F}_{max}$ | Maximum working load | 1000.0 (lb) |

S | Maximum allowable shear stress | $1.89\times {10}^{5}$ (psi) |

${l}_{max}$ | Maximum free length | 14.0 (in) |

${d}_{min}$ | Minimum wire diameter | 0.2 (in) |

${D}_{max}$ | Maximum outside spring diameter | 3.0 (in) |

${F}_{p}$ | Preload compression force | 300.0 (lb) |

${\delta}_{pm}$ | Maximum allowable deflection | 6.0 (in) |

under preload | ||

${\delta}_{w}$ | Deflection from preload position to | 1.25 (in) |

maximum load position | ||

G | Shear modulus of the material | $1.15\times {10}^{7}$ (psi) |

Reference Method | [40] NLPA ^{a} | [38] GA | [39] HSIA | [21] DE | [14] PSO | [1] FA | [2] PSO | Present Study PSO (${\mathit{n}}_{\mathit{s}}=16$) |
---|---|---|---|---|---|---|---|---|

${x}_{1}$ (D) | 1.180701 | 1.227411 | 1.223 | 1.223041 | 1.223041 | 1.223049 | 1.223041 | 1.223041 |

${x}_{2}$ (N) | 10 | 9 | 9 | 9 | 9 | 9 | 9 | 9 |

${x}_{3}$ (d) | 0.283 | 0.283 | 0.283 | 0.283 | 0.283 | 0.283 | 0.283 | 0.283 |

$-{h}_{1}\left({\mathit{x}}^{*}\right)$ | 5430.9 | 550.993 | 1008.81 | 1008.8114 | 1008.8114 | 1008.02 | 1008.8059 | 1008.811398 |

$-{h}_{2}\left({\mathit{x}}^{*}\right)$ | 8.8187 | 8.9264 | 8.946 | 8.94564 | 8.9456 | 8.946 | 8.945635 | 8.945636 |

$-{h}_{3}\left({\mathit{x}}^{*}\right)$ | 0.08298 | 0.0830 | 0.083 | 0.08300 | 0.083 | 0.083 | 0.083000 | 0.083000 |

$-{h}_{4}\left({\mathit{x}}^{*}\right)$ | 1.8193 | 1.7726 | 1.77696 | 1.77696 | 1.777 | 1.777 | 1.493959 | 1.493959 |

$-{h}_{5}\left({\mathit{x}}^{*}\right)$ | 1.1723 | 1.3371 | 1.32170 | 1.32170 | 1.3217 | 1.322 | 1.321700 | 1.321700 |

$-{h}_{6}\left({\mathit{x}}^{*}\right)$ | 5.4643 | 5.4585 | 5.4643 | 5.46429 | 5.4643 | 5.464 | 5.464286 | 5.464286 |

$-{h}_{7}\left({\mathit{x}}^{*}\right)$ | 0.0 | 0.0 | 0.0 | 0.0 | 0.0000 | 0 | 0.000000 | 0.000000 |

$-{h}_{8}\left({\mathit{x}}^{*}\right)$ | 0.0 | 0.0134 | 0.0 | 0.0 | 0.0000 | 0.0000 | 0.000010 | 0.000000 |

Best objective value | 2.7995 | 2.6681 | 2.659 | 2.65856 | 2.65856 | 2.658576 | 2.658559 | 2.658559 |

Objective deviation | $1.0+5.301\times {10}^{-2}$ | $1.0+3.589\times {10}^{-3}$ | $1.0+1.659\times {10}^{-4}$ | $1.0+3.761\times {10}^{-7}$ | $1.0+3.761\times {10}^{-7}$ | $1.0+6.394\times {10}^{-6}$ | 1.0 | 1.0 |

Worst objective value | N/A | N/A | N/A | N/A | N/A | 7.8162919 | N/A | 2.660784 |

Average objective value | N/A | N/A | N/A | N/A | 2.738024 | 4.3835958 | N/A | 2.658890 |

Standard deviation | N/A | N/A | N/A | N/A | N/A | 4.6076313 | N/A | 0.000611 |

Function evaluations | N/A | N/A | N/A | 26,000 | 15,000 | 75,000 | 4784–98,992 | 50,000 |

^{a}Nonlinear programming algorithm.

a | ≤1.4 | 1.5 | 1.6 | 1.7 | 1.8 | 1.9 | 2 | 2.1 | 2.2 | 2.3 | 2.4 | 2.5 | 2.6 | 2.7 | ≥2.8 |

$g\left(a\right)$ | 0.58 | 1 | 0.85 | 0.77 | 0.71 | 0.66 | 0.63 | 0.6 | 0.56 | 0.55 | 0.53 | 0.52 | 0.51 | 0.51 | 0.5 |

Reference Method | [42] | [43] | [41] OPTIVAR | [3] NBV | [30] | Present Study PSO (${\mathit{n}}_{\mathit{s}}=16$) | ||
---|---|---|---|---|---|---|---|---|

GeneAS-I | GeneAS-II | ABC | TLBO | |||||

${x}_{1}$ (t) | 0.208 | 0.205 | 0.210 | 0.204 | 0.204143 | N/A | 0.204143 | 0.204143 |

${x}_{2}$ (h) | 0.2 | 0.201 | 0.204 | 0.200 | 0.2 | N/A | 0.2 | 0.200000 |

${x}_{3}$ (${D}_{i}$) | 8.751 | 9.534 | 9.268 | 10.030 | 10.0304732 | N/A | 10.03047 | 10.030473 |

${x}_{4}$ (${D}_{e}$) | 11.067 | 11.627 | 11.499 | 12.010 | 12.01 | N/A | 12.01 | 12.010000 |

${h}_{1}\left({\mathit{x}}^{*}\right)$ | 2145.4109 | −10.3396 | 2127.2624 | 134.0816 | $4.58\times {10}^{-4}$ | N/A | $1.77\times {10}^{-6}$ | $6.93\times {10}^{-11}$ |

${h}_{2}\left({\mathit{x}}^{*}\right)$ | 39.75018 | 2.8062 | 194.222554 | −12.5328 | $3.04\times {10}^{-7}$ | N/A | $7.46\times {10}^{-8}$ | $8.85\times {10}^{-10}$ |

${h}_{3}\left({\mathit{x}}^{*}\right)$ | 0.00000 | 0.0010 | 0.0040 | 0.0000 | $9.24\times {10}^{-10}$ | N/A | $5.8\times {10}^{-11}$ | $8.81\times {10}^{-13}$ |

${h}_{4}\left({\mathit{x}}^{*}\right)$ | 1.592 | 1.5940 | 1.5860 | 1.5960 | 1.595856 | N/A | 1.595857 | 1.595857 |

${h}_{5}\left({\mathit{x}}^{*}\right)$ | 0.943 | 0.3830 | 0.5110 | 0.0000 | 0 | N/A | $2.35\times {10}^{-9}$ | $1.58\times {10}^{-10}$ |

${h}_{6}\left({\mathit{x}}^{*}\right)$ | 2.316 | 2.0930 | 2.2310 | 1.9800 | 1.979526 | N/A | 1.979527 | 1.979527 |

${h}_{7}\left({\mathit{x}}^{*}\right)$ | 0.21364 | 0.20397 | 0.20856 | 0.19899 | 0.198965 | N/A | 0.198966 | 0.198966 |

Best objective value | 2.121964 | 2.01807 | 2.16256 | 1.978715 | 1.979675 | 1.979675 | 1.979675 | 1.979675 |

Objective deviation | $1.0+7.187\times {10}^{-2}$ | $1.0+1.939\times {10}^{-2}$ | $1.0+9.238\times {10}^{-2}$ | $1.0-4.849\times {10}^{-4}$ | 1.0 | 1.0 | 1.0 | 1.0 |

Feasibility | Feasible | Infeasible | Feasible | Infeasible | Feasible | N/A | Feasible | Feasible |

Worst objective value | N/A | N/A | N/A | N/A | 2.005431 | 2.104297 | 1.979757 | 1.979675 |

Average objective value | N/A | N/A | N/A | N/A | 1.984698 | 1.995475 | 1.979688 | 1.979675 |

Standard deviation | N/A | N/A | N/A | N/A | 7.78 $\times {10}^{-3}$ | 0.07 | 0.45 | 0.000000 |

Function evaluations | N/A | N/A | N/A | N/A | 15,000 | 150,000 | 150,000 | 50,000 |

Reference Method | DEDS | DELC | HEAA | MDE | PSO-DE | MBA | Present Study PSO (${\mathit{n}}_{\mathit{s}}=16$) |
---|---|---|---|---|---|---|---|

${x}_{1}$ (b) | 3.5 | 3.5 | 3.500022 | 3.500010 | 3.5000000 | 3.500000 | 3.500000 |

${x}_{2}$ (m) | 0.7 | 0.7 | 0.70000039 | 0.70000 | 0.700000 | 0.700000 | 0.700000 |

${x}_{3}$ (n) | 17 | 17 | 17.000012 | 17 | 17.000000 | 17.000000 | 17.000000 |

${x}_{4}$ (${l}_{1}$) | 7.3 | 7.3 | 7.300427 | 7.300156 | 7.300000 | 7.300033 | 7.300000 |

${x}_{5}$ (${l}_{2}$) | 7.715319 | 7.715319 | 7.715377 | 7.800027 | 7.800000 | 7.715772 | 7.800000 |

${x}_{6}$ (${d}_{1}$) | 3.350214 | 3.350214 | 3.350230 | 3.350221 | 3.350214 | 3.350218 | 2.900000 |

${x}_{7}$ (${d}_{2}$) | 5.286654 | 5.286654 | 5.286663 | 5.286685 | 5.2866832 | 5.286654 | 5.286683 |

Best objective value | 2994.471066 | 2994.471066 | 2994.499107 | 2996.356689 | 2996.348167 | 2994.482453 | 2896.259285 |

Objective deviation | $1.0+3.3910\times {10}^{-2}$ | $1.0+3.3910\times {10}^{-2}$ | $1.0+3.392\times {10}^{-2}$ | $1.0+3.4561\times {10}^{-2}$ | $1.0+3.4558\times {10}^{-2}$ | $1.0+3.3914\times {10}^{-2}$ | 1.0 |

Worst objective value | 2994.471066 | 2994.471066 | 2994.752311 | 2996.390137 | 2996.348204 | 2999.652444 | 2896.259380 |

Average objective value | 2994.471066 | 2994.471066 | 2994.613368 | 2996.367220 | 2996.348174 | 2996.769019 | 2896.259292 |

Standard deviation | $3.6\times {10}^{-12}$ | $1.9\times {10}^{-12}$ | $7.0\times {10}^{-2}$ | $8.2\times {10}^{-3}$ | $6.4\times {10}^{-6}$ | 1.56 | 0.000017 |

Function evaluations | 30,000 | 30,000 | 40,000 | 24,000 | 54,350 | 25,000 | 50,000 |

Method | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | ${\mathit{x}}_{4}$ | ${\mathit{x}}_{5}$ | ${\mathit{x}}_{6}$ | ${\mathit{x}}_{7}$ | ${\mathit{x}}_{8}$ | ${\mathit{x}}_{9}$ | ${\mathit{x}}_{10}$ | Objective Function | Function | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(${\mathit{b}}_{\mathbf{1}}$) | (${\mathit{h}}_{\mathbf{1}}$) | (${\mathit{b}}_{\mathbf{2}}$) | (${\mathit{h}}_{\mathbf{2}}$) | (${\mathit{b}}_{\mathbf{3}}$) | (${\mathit{h}}_{\mathbf{3}}$) | (${\mathit{b}}_{\mathbf{4}}$) | (${\mathit{h}}_{\mathbf{4}}$) | (${\mathit{b}}_{\mathbf{5}}$) | (${\mathit{h}}_{\mathbf{5}}$) | Best | Deviation | Worst | Mean | Std. Dev. | Evaluations | ||

[46] | RNES 1 ^{a} | 3 | 60 | 3.1 | 55 | 2.6 | 50 | 2.311 | 43.108 | 1.822 | 34.307 | 64269.594 | $1.0+5.887\times {10}^{-3}$ | N/A | 12,000 | ||

RNES 2 | 3 | 60 | 3.1 | 55 | 2.6 | 50 | 2.267 | 43.797 | 1.849 | 34.282 | 64322.433 | $1.0+6.714\times {10}^{-3}$ | N/A | 12,000 | |||

RNES 3 | 3 | 60 | 3.1 | 55 | 2.6 | 50 | 2.348 | 42.804 | 1.783 | 34.753 | 64299.108 | $1.0+6.349\times {10}^{-3}$ | N/A | 12,000 | |||

RNES 4 | 3 | 60 | 3.1 | 55 | 2.6 | 50 | 2.491 | 41.51 | 2.113 | 33.231 | 65416.896 | $1.0+2.384\times {10}^{-2}$ | N/A | 12,000 | |||

[47] | DOT | N/A | 65391.59 | $1.0+2.345\times {10}^{-2}$ | N/A | N/A | |||||||||||

SLP ^{b} | N/A | 65451.50 | $1.0+2.439\times {10}^{-2}$ | N/A | N/A | ||||||||||||

MLD ^{c}-SLP | N/A | 65352.20 | $1.0+2.283\times {10}^{-2}$ | N/A | N/A | ||||||||||||

[48] | C/RU ^{d} | 4 | 62 | 3.1 | 60 | 2.6 | 55 | 2.205 | 44.09 | 1.751 | 35.03 | 73555 | $1.0+1.512\times {10}^{-1}$ | N/A | N/A | ||

PD ^{e} | 3 | 60 | 3.1 | 55 | 2.6 | 50 | 2.276 | 45.528 | 1.75 | 34.995 | 64537 | $1.0+1.007\times {10}^{-2}$ | N/A | N/A | |||

LAD ^{f} | 3 | 60 | 3.1 | 55 | 2.6 | 50 | 2.262 | 45.233 | 1.75 | 34.995 | 64403 | $1.0+7.975\times {10}^{-3}$ | N/A | N/A | |||

CAD ^{g} | 3 | 60 | 3.1 | 55 | 2.6 | 50 | 2.279 | 45.553 | 1.75 | 35.004 | 64558 | $1.0+1.040\times {10}^{-2}$ | N/A | N/A | |||

[49] | GAOS Level 1 | 3 | 60 | 3.1 | 55 | 2.6 | 50 | 2.3 | 45.5 | 1.8 | 35 | 64815 | $1.0+1.442\times {10}^{-2}$ | N/A | 10,000 | ||

GAOS Level 2 | 3 | 60 | 3.1 | 55 | 2.6 | 50 | 2.27 | 45.25 | 1.75 | 35 | 64447 | $1.0+8.664\times {10}^{-3}$ | N/A | 10,000 | |||

[50] | GA-APM ^{h} | 3 | 60 | 3.1 | 55 | 2.6 | 50 | 2.2894 | 45.6256 | 1.7931 | 34.593 | 64698.56 | $1.0+1.260\times {10}^{-2}$ | 73931.359 | 68107.046 | N/A | 35,000 |

[51] | AIS-GA | 3 | 60 | 3.1 | 55 | 2.6 | 50 | 2.235 | 44.395 | 2.004 | 32.879 | 65559.60 | $1.0+2.608\times {10}^{-2}$ | 77272.78 | 70857.12 | N/A | 35,000 |

AIS-GA-C ^{i} | 3 | 60 | 3.1 | 60 | 2.6 | 50 | 2.311 | 43.186 | 2.225 | 31.250 | 66533.47 | $1.0+4.132\times {10}^{-2}$ | 76852.86 | 71821.69 | N/A | 35,000 | |

[1] | FA | 3 | 60 | 3.1 | 55 | 2.6 | 50 | 2.205 | 44.091 | 1.750 | 34.995 | 63893.52 | $1.0+1.409\times {10}^{-6}$ | 64262.99420 | 64144.75312 | 175.91879 | 50,000 |

Present study | PSO (${n}_{s}=16$) | 3 | 60 | 3.1 | 55 | 2.6 | 50 | 2.20456 | 44.09111 | 1.74976 | 34.99514 | 63893.43 | 1.0 | 63893.43080 | 63893.43080 | 0.00000 | 50,000 |

^{a}Rank-niche evolution strategy.

^{b}Sequential linear programming.

^{c}Move limit definition.

^{d}Continuous/round up.

^{e}Precise discrete.

^{f}Linear approximate discrete.

^{g}Conservative approximate discrete.

^{h}Adaptive penalty method.

^{i}Clearing.

Reference Method | [52] GA | [3] MBA | [30] | Present Study PSO (n_{s} = 16) | |
---|---|---|---|---|---|

ABC | TLBO | ||||

${x}_{1}$ (${D}_{b}$) | 125.7171 | 125.7153 | N/A | 125.7191 | 125.719056 |

${x}_{2}$ (${D}_{m}$) | 21.423 | 21.423300 | N/A | 21.42559 | 21.425590 |

${x}_{3}$ (Z) | 11 | 11.000 | N/A | 11 | 11.000000 |

${x}_{4}$ (${f}_{i}$) | 0.515 | 0.515000 | N/A | 0.515 | 0.515000 |

${x}_{5}$ (${f}_{o}$) | 0.515 | 0.515000 | N/A | 0.515 | 0.515000 |

${x}_{6}$ (${K}_{Dmin}$) | 0.4159 | 0.488805 | N/A | 0.424266 | 0.411776 |

${x}_{7}$ (${K}_{Dmax}$) | 0.651 | 0.627829 | N/A | 0.633948 | 0.613510 |

${x}_{8}$ ($\epsilon $) | 0.300043 | 0.300149 | N/A | 0.3 | 0.300000 |

${x}_{9}$ (e) | 0.0223 | 0.097305 | N/A | 0.068858 | 0.059359 |

${x}_{10}$ ($\zeta $) | 0.751 | 0.646095 | N/A | 0.799498 | 0.667473 |

${h}_{1}\left({\mathit{x}}^{*}\right)$ | 0.000821 | 0 | N/A | 0 ^{a} | 0.000000 |

${h}_{2}\left({\mathit{x}}^{*}\right)$ | 13.732999 | 8.630183 | N/A | 13.15257 | 14.026828 |

${h}_{3}\left({\mathit{x}}^{*}\right)$ | 2.724000 | 1.101429 | N/A | 1.5252 | 0.094509 |

${h}_{4}\left({\mathit{x}}^{*}\right)$ | −1.107 | 2.040448 | N/A | 0.719056 | −1.401405 |

${h}_{5}\left({\mathit{x}}^{*}\right)$ | 0.717000 | 0.715366 | N/A | 16.49544 | 0.719056 |

${h}_{6}\left({\mathit{x}}^{*}\right)$ | 4.857899 | 23.611002 | N/A | 0 | 14.120649 |

${h}_{7}\left({\mathit{x}}^{*}\right)$ | 0.0021 | 0.000480 | N/A | 0 | 0.000000 |

${h}_{8}\left({\mathit{x}}^{*}\right)$ | 0.000007 | 0 | N/A | 2.559363 | 0.000000 |

${h}_{9}\left({\mathit{x}}^{*}\right)$ | 0.000007 | 0 | N/A | 0 | 0.000000 |

Best objective value | 81,843.3 | 85,535.9611 ^{b} | 81,859.7416 | 81,859.74 | 81,859.7416 |

Objective deviation | $1.0-2.009\times {10}^{-4}$ | $1.0+4.491\times {10}^{-2}$ | 1.0 | $1.0-1.955\times {10}^{-8}$ | 1.0 |

Feasibility | Feasible | Infeasible | N/A | Infeasible | Feasible |

Worst objective value | N/A | 84,440.1948 | 78,897.81 | 80,807.8551 | 81,859.7401 |

Average objective value | N/A | 85,321.4030 | 81,496 | 81,438.987 | 81,859.7415 |

Standard deviation | N/A | 211.52 | 0.69 | 0.66 | 0.0003 |

Function evaluations | 225,000 | 50,000 | 10,000 | 10,000 | 50,000 |

^{a}Some constraint function values do not coincide with those calculated using the provided optimal solution. Their solution provides ${h}_{1}\left({\mathit{x}}^{*}\right)=0.000004$, ${h}_{2}\left({\mathit{x}}^{*}\right)=13.152560$, ${h}_{3}\left({\mathit{x}}^{*}\right)=1.525180$, ${h}_{4}\left({\mathit{x}}^{*}\right)=2.559350$, ${h}_{5}\left({\mathit{x}}^{*}\right)=0.719100$, ${h}_{6}\left({\mathit{x}}^{*}\right)=16.495400$, ${h}_{7}\left({\mathit{x}}^{*}\right)=-0.000022$, ${h}_{8}\left({\mathit{x}}^{*}\right)=0.0$, and ${h}_{9}\left({\mathit{x}}^{*}\right)=0.0$.

^{b}This best value does not coincide with that calculated using the provided optimal solution. Their solution provides $f\left({\mathit{x}}^{*}\right)=81843.68625$.

Reference Method | [1] | Present Study PSO (${\mathit{n}}_{\mathit{s}}=16$) | |||
---|---|---|---|---|---|

PSO | DE | GA | FA | ||

${x}_{1}$ | 0.50000 | 0.50000 | 0.50005 | 0.50000 | 0.500000 |

${x}_{2}$ | 1.11670 | 1.11670 | 1.28017 | 1.36000 | 1.116366 |

${x}_{3}$ | 0.50000 | 0.5000 | 0.50001 | 0.50000 | 0.500000 |

${x}_{4}$ | 1.30208 | 1.30208 | 1.03302 | 1.20200 | 1.302197 |

${x}_{5}$ | 0.50000 | 0.50000 | 0.50001 | 0.50000 | 0.500000 |

${x}_{6}$ | 1.50000 | 1.50000 | 0.50000 | 1.12000 | 1.500000 |

${x}_{7}$ | 0.50000 | 0.50000 | 0.50000 | 0.50000 | 0.500000 |

${x}_{8}$ | 0.34500 | 0.34500 | 0.34994 ^{a} | 0.34500 | 0.345000 |

${x}_{9}$ | 0.19200 | 0.19200 | 0.19200 | 0.19200 | 0.192000 |

${x}_{10}$ | −19.54935 | −19.54935 | 10.3119 | 8.87307 | −19.561544 |

${x}_{11}$ | −0.00431 | −0.00431 | 0.00167 | −18.99808 | −0.000190 |

Best objective value | 22.84474 | 22.84298 ^{b} | 22.85653 | 22.84298 ^{c} | 22.842969 |

Objective deviation | $1.0+7.753\times {10}^{-5}$ | $1.0+4.815\times {10}^{-7}$ | $1.0+5.937\times {10}^{-4}$ | $1.0+4.815\times {10}^{-7}$ | 1.0 |

Feasibility | Feasible | Feasible | Infeasible | Infeasible | Feasible |

Worst objective value | 23.21354 | 24.12606 | 26.240578 | 24.06623 | 22.846465 |

Average objective value | 22.89429 | 23.22828 | 23.51585 | 22.89376 | 22.843136 |

Standard deviation | 0.15017 | 0.34451 | 0.66555 | 0.16667 | 0.000649 |

^{a}${x}_{8}\notin \{0.192,0.345\}.$

^{b}The optimal solution is identical to that of PSO but different objective function values are provided in Gandomi et al. [1].

^{c}This best value does not coincide with that calculated using the provided optimal solution, $f\left({\mathit{x}}^{*}\right)=24.06622$. Furthermore, their optimal solution may not guarantee the satisfaction of two constraint conditions, ${h}_{8}\left({\mathit{x}}^{*}\right)$ = 4.02129 > 4 kN and ${h}_{10}\left(\mathit{x}\right)$ = 15.84839 > 15.7 mm/ms.

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**MDPI and ACS Style**

Kim, T.-H.; Cho, M.; Shin, S.
Constrained Mixed-Variable Design Optimization Based on Particle Swarm Optimizer with a Diversity Classifier for Cyclically Neighboring Subpopulations. *Mathematics* **2020**, *8*, 2016.
https://doi.org/10.3390/math8112016

**AMA Style**

Kim T-H, Cho M, Shin S.
Constrained Mixed-Variable Design Optimization Based on Particle Swarm Optimizer with a Diversity Classifier for Cyclically Neighboring Subpopulations. *Mathematics*. 2020; 8(11):2016.
https://doi.org/10.3390/math8112016

**Chicago/Turabian Style**

Kim, Tae-Hyoung, Minhaeng Cho, and Sangwoo Shin.
2020. "Constrained Mixed-Variable Design Optimization Based on Particle Swarm Optimizer with a Diversity Classifier for Cyclically Neighboring Subpopulations" *Mathematics* 8, no. 11: 2016.
https://doi.org/10.3390/math8112016