# OPTIMUS: Self-Adaptive Differential Evolution with Ensemble of Mutation Strategies for Grasshopper Algorithmic Modeling

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## Abstract

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## 1. Introduction

#### 1.1. The Necessity of Optimization in Architecture

#### 1.2. Performative Computational Architecture Framework

#### 1.3. Current Optimization Tools in Grasshopper

#### 1.3.1. Galapagos

#### 1.3.2. Goat

#### 1.3.3. Silvereye

#### 1.3.4. Opossum

#### 1.3.5. Dodo

#### 1.3.6. Nelder–Mead Optimization

#### 1.4. This Study: Optimus

- Supports PCA framework outlined in previous section.
- Presents the highest performance when compared to other optimization algorithms available in GH reviewed in Section 1.3.
- ○
- The performance of the Optimus is tested by benchmark suite, which consists of standard single objective unconstrained problems and some of the test problems proposed in IEEE Congress on Evolutionary Computation 2005 (CEC 2005) [44]. Problem formulations and optimization results of these benchmarks are given in Section 4.1.
- ○
- Finally, Optimus is tested with a design optimization problem. The problem formulation and optimization results are given in Section 4.2.

## 2. Self-Adaptive Differential Evolution Algorithm with Ensemble of Mutation Strategies

#### 2.1. The Basic Differential Evolution

- Initialization for generating the initial target population once at the beginning.
- Reproduction with mutation for generating the mutant population by using the target population.
- Reproduction with crossover for generating the trial population by using the mutant population.
- Selection to choose the next generation among trial and target populations using one-to-one comparison. In each generation, individuals of the current population become the target population.

#### 2.1.1. Initialization

#### 2.1.2. Mutation

#### 2.1.3. Crossover

#### 2.1.4. Selection

#### 2.2. Self-Adaptive Approach

#### 2.3. Ensemble Approach

Algorithm 1. The self-adaptive differential evolution algorithm with ensemble of mutation strategies. |

1: Set parameters g = 0, NP = 100, ${M}_{max}$ = 4 2: Establish initial population randomly 3: ${P}^{g}=\left\{{x}_{1}^{g},..,{x}_{NP}^{g}\right\}\text{}with\text{}{x}_{i}^{g}=\left\{{x}_{i1}^{g},..,{x}_{iD}^{g}\right\}$ 4: Assign a mutation strategy to each individual randomly 5: ${M}_{i}=rand()\%{M}_{max}$ $for\text{}i=1,..,NP$ 6: Evaluate population and find ${x}_{best}^{g}$ 7: $f\left({P}^{g}\right)=\left\{f\left({x}_{1}^{g}\right),..,f\left({x}_{NP}^{g}\right)\right\}$ 8: Assign CR[i] = 0.5 and F[i] = 0.9 to each individual 9: Repeat the following for each individual ${x}_{i}^{g}$ 10: Obtain ${v}_{i}^{g}={M}_{i}\left({x}_{i}^{g}\right)$ 11: Obtain ${u}_{i}^{g}=C{R}_{i}\left({x}_{i}^{g},{v}_{i}^{g}\right)$ 12: Obtain ${x}_{i}^{g}=\{\begin{array}{cc}{u}_{i}^{g}& if\text{}f\left({u}_{i}^{g}\right)\le f\left({x}_{i}^{g-1}\right)\\ {x}_{i}^{g-1}& otherwise\end{array}$ 13: If $f\left({u}_{i}^{g}\right)>f\left({x}_{i}^{g-1}\right)$, ${M}_{i}=rand()\%{M}_{max}$ 14: If $f\left({x}_{i}^{g}\right)\le f\left({x}_{best}^{g}\right)$, ${x}_{best}^{g}={x}_{i}^{g}$ 15: Update ${F}_{i}^{g}$ and $C{R}_{i}^{g}$ 16: If the stopping criterion is not met, go to Lines 9–15 17: Else stop and return ${\pi}_{best}$ |

## 3. Optimus

- Optimus consists of
- ReadMe (giving details about Optimus),
- jEDE (using jEDE algorithm, generating mutant and trial populations)
- One-to-one (enabling selection process for next generation)

- Initialize consists of
- GetBound (taking the boundaries of design variables in $D$ dimensions)
- InitPop (generating initial population for $NP$ population size in $D$ dimensions)

- Utilities consists of
- xBest (finding chromosomes that has the lowest fitness value in the population)
- Merge (collecting the fitness, population, xBest and optimization parameters)
- Unmerge (separating the fitness, population, xBest and optimization parameters)

- Place GetBound on the GH canvas and connect with number sliders.
- Define the population size.
- Get InitPop for initialization using population size and output of GetBound.
- Evaluate initial fitness using the output of InitPop.
- Internalize the initial fitness.
- Place xBest on the GH canvas.
- Get Merge and connect with internalized initial fitness and outputs of InitPop and xBest.
- Connect Merge with starting input (S) of HoopSnake.
- Place UnMerge on the GH canvas and connect with feedback output (F) of HoopSnake.
- Get jEDE and connect outputs of UnMerge, InitPop, GetBound.
- Evaluate trial fitness using the output of jEDE.
- Get One-to-One and connect with initial fitness, trial fitness and outputs of jEDE.
- Place xBest and connect outputs of One-to-one for updating the new best chromosome.
- Get Merge and connect outputs of jEDE, One-to-one, and xBest.
- Complete the loop by connecting the output of Merge to the data input (D) of the HoopSnake.

## 4. Experiments

#### 4.1. Benchmark Suite

#### 4.1.1. Experimental Setup and Evaluation Criteria

#### 4.1.2. Experimental Results

#### 4.2. Design Optimization Problem

#### 4.2.1. Experimental Setup and Evaluation Criteria

#### 4.2.2. Design Results

## 5. Discussion

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

${F}_{sph}$: Sphere Function${F}_{sph}\left(x\right)={\displaystyle \sum _{i=1}^{D}{x}_{i}^{2}}$ ${F}_{sph}^{\ast}\left({x}^{\ast}\right)=\left(0,..,0\right)=0$$-100\le x\le 100$ |

${F}_{ros}$: Rosenbrock’s Function${F}_{ros}\left(x\right)={\displaystyle \sum _{i=1}^{D-1}\left(100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left(1-{x}_{i}\right)}^{2}\right)}$ ${F}_{ros}^{\ast}\left({x}^{\ast}\right)=\left(1,..,1\right)=0$$-100\le x\le 100$ |

${F}_{ack}$: Ackley’s Function${F}_{ack}\left(x\right)=-20\mathrm{exp}\left(-0.2\sqrt{\frac{1}{D}{\displaystyle \sum _{i=1}^{D}{x}_{i}^{2}}}\right)-\mathrm{exp}\left(\frac{1}{D}{\displaystyle \sum _{i=1}^{D}\mathrm{cos}\left(2\pi {x}_{i}\right)}\right)+20+e$ ${F}_{ack}^{\ast}\left({x}^{\ast}\right)=\left(0,..,0\right)=0$$-32\le x\le 32$ |

${F}_{grw}$: Griewank’s Function${F}_{grw}\left(x\right)={\displaystyle \sum _{i=1}^{D}\frac{{x}_{i}^{2}}{4000}-{\displaystyle \prod _{i=1}^{D}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1}}$ ${F}_{grw}^{\ast}\left({x}^{\ast}\right)=\left(0,..,0\right)=0$$-600\le x\le 600$ |

${F}_{ras}$: Rastrigin’s Function ${F}_{ras}\left(x\right)={\displaystyle \sum _{i=1}^{D}\left({x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)+10\right)}$ ${F}_{ras}^{\ast}\left({x}^{\ast}\right)=\left(0,..,0\right)=0$$-5\le x\le 5$ |

${F}_{sch}$: Generalized Schwefel’s Problem 2.26 ${F}_{sch}\left(x\right)=418.9829N-{\displaystyle \sum _{i=1}^{D}\left({x}_{i}\mathrm{sin}\left(\left|{x}_{i}\right|\right)\right)}$ ${F}_{sch}^{\ast}\left({x}^{\ast}\right)=\left(420.9687,..,420.968\right)=0$$-500\le x\le 500$ |

${F}_{sal}$: Salomon’s Function ${F}_{sal}\left(x\right)=-\mathrm{cos}\left(2\pi \sqrt{{\displaystyle \sum _{i=1}^{D}{x}_{i}^{2}}}\right)+0.1\sqrt{{\displaystyle \sum _{i=1}^{D}{x}_{i}^{2}}}+1$ ${F}_{sal}^{\ast}\left({x}^{\ast}\right)=\left(0,..,0\right)=0$$-100\le x\le 100$ |

${F}_{wht}$: Whitely’s Function ${F}_{wht}\left(x\right)={\displaystyle \sum _{j=1}^{D}{\displaystyle \sum _{i=1}^{D}\left(\frac{{y}_{ij}^{2}}{4000}-\mathrm{cos}\left({y}_{ij}\right)+1\right)}}$ where ${y}_{ij}=100{\left({x}_{j}-{x}_{i}\right)}^{2}+{\left(1-{x}_{i}\right)}^{2}$ ${F}_{wht}^{\ast}\left({x}^{\ast}\right)=\left(1,..,1\right)=0$$-100\le x\le 100$ |

${F}_{pn1}$: Generalized Penalized Function 1 ${F}_{pn1}\left(x\right)=\frac{\pi}{D}\left\{10{\mathrm{sin}}^{2}\left(\pi {y}_{1}\right)+{\displaystyle \sum _{i=1}^{D-1}{\left({y}_{i}-1\right)}^{2}\left[1+10{\mathrm{sin}}^{2}\left(\pi {y}_{i+1}\right)\right]}+{\left({y}_{D}-1\right)}^{2}\right\}+{\displaystyle \sum _{i=1}^{D}u\left({x}_{i},10,100,4\right)}$ where ${y}_{i}=1+\frac{1}{4}\left({x}_{i}+1\right)$ and $u\left({x}_{i},a,k,m\right)=\{\begin{array}{c}\begin{array}{cc}k{\left({x}_{i}-a\right)}^{m}& {x}_{i}>a\end{array}\\ \begin{array}{cc}0& -a\le {x}_{i}\le a\end{array}\\ \begin{array}{cc}k{\left(-{x}_{i}-a\right)}^{m}& {x}_{i}<a\end{array}\end{array}$ ${F}_{pn1}^{\ast}\left({x}^{\ast}\right)=\left(-1,..,-1\right)=0$ $-50\le x\le 50$ |

${F}_{pn2}$: Generalized Penalized Function 2 $\begin{array}{l}{F}_{pn2}\left({x}^{\ast}\right)=0.1\left\{{\mathrm{sin}}^{2}\left(3\pi {x}_{1}\right)+{\displaystyle \sum _{i=1}^{D-1}{\left({x}_{i-1}-1\right)}^{2}}\left[1+{\mathrm{sin}}^{2}\left(3\pi {x}_{i+1}\right)\right]+{\left({x}_{D}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}\left(2\pi {x}_{D}\right)\right]\right\}+\\ {\displaystyle \sum _{i=1}^{D}u\left({x}_{i},5,100,4\right)}\end{array}$ ${F}_{pn1}^{\ast}\left({x}^{\ast}\right)=\left(1,..,1\right)=0$$-50\le x\le 50$ |

${F}_{1}$: Shifted Sphere Function ${F}_{1}\left(x\right)={\displaystyle \sum _{i=1}^{D}{z}_{i}^{2}+f\_bia{s}_{1}}$ $x\in {\left[-100,100\right]}^{D}$, ${F}_{1}\left({x}^{\ast}\right)=f\_bia{s}_{1}=-450$ |

${F}_{2}$: Shifted Schwefel’s Problem 1.2 ${F}_{2}\left(x\right)={\displaystyle \sum _{i=1}^{D}{\left({\displaystyle \sum _{j=1}^{i}{z}_{j}}\right)}^{2}}+f\_bia{s}_{2}$ $x\in {\left[-100,100\right]}^{D}$, ${F}_{2}\left({x}^{\ast}\right)=f\_bia{s}_{2}=-450$ |

${F}_{3}$: Shifted Rotated High Conditioned Elliptic Function ${F}_{3}\left(x\right)={\displaystyle \sum _{i=1}^{D}{\left({10}^{6}\right)}^{\frac{i-1}{D-1}}{z}_{i}^{2}+f\_bia{s}_{3}}$ $x\in {\left[-100,100\right]}^{D}$, ${F}_{3}\left({x}^{\ast}\right)=f\_bia{s}_{3}=-450$ |

${F}_{4}$: Shifted Schwefel’s Problem 1.2 With Noise in Fitness ${F}_{4}\left(x\right)=\left({\displaystyle \sum _{i=1}^{D}{\left({\displaystyle \sum _{j=1}^{i}{z}_{i}}\right)}^{2}}\right)\times \left(1+0.4\left|N\left(0,1\right)\right|\right)+f\_bia{s}_{4}$ $x\in {\left[-100,100\right]}^{D}$, ${F}_{4}\left({x}^{\ast}\right)=f\_bia{s}_{4}=-450$ |

${F}_{5}$: Schwefel’s Problem 2.6 with Global Optimum on Bounds ${F}_{5}\left(x\right)=\mathrm{max}\left\{\left|{A}_{i}x-{B}_{i}\right|\right\}+f\_bia{s}_{5},i=1,2,..,D$ $x\in {\left[-100,100\right]}^{D}$, ${F}_{5}\left({x}^{\ast}\right)=f\_bia{s}_{5}=-310$ |

${F}_{6}$: Shifted Rosenbrock’s Function ${F}_{6}\left(x\right)={\displaystyle \sum _{i=1}^{D-1}\left(100{\left({z}_{i}^{2}-{z}_{i+1}\right)}^{2}+{\left({z}_{i}-1\right)}^{2}\right)+f\_bia{s}_{6}}$ $x\in {\left[-100,100\right]}^{D}$, ${F}_{4}\left({x}^{\ast}\right)=f\_bia{s}_{6}=390$ |

${F}_{7}$: Shifted Rotated Griewank’s Function without Bounds ${F}_{7}\left(x\right)={\displaystyle \sum _{i=1}^{D}\frac{{z}_{i}^{2}}{4000}-{\displaystyle \prod _{i=1}^{D}\mathrm{cos}\left(\frac{{z}_{i}}{\sqrt{i}}\right)+1+f\_bia{s}_{7}}}$ Initialize population in ${\left[0,600\right]}^{D}$, Global optimum is outside of the initialization range, ${F}_{7}\left({x}^{\ast}\right)=f\_bia{s}_{7}=-180$ |

${F}_{8}$: Shifted Rotated Ackley’s Function with Global Optimum on Bounds. ${F}_{8}\left(x\right)=-20\mathrm{exp}\left(-0.2\sqrt{\frac{1}{D}{\displaystyle \sum _{i=1}^{D}{z}_{i}^{2}}}\right)-\mathrm{exp}\left(\frac{1}{D}{\displaystyle \sum _{i=1}^{D}\mathrm{cos}\left(2\pi z\right)}\right)+20+e+f\_bia{s}_{8}$ $x\in {\left[-32,32\right]}^{D}$, ${F}_{8}\left({x}^{\ast}\right)=f\_bia{s}_{8}=-140$ |

${F}_{9}$: Shifted Rastrigin’s Function ${F}_{9}\left(x\right)={\displaystyle \sum _{i=1}^{D}\left({z}_{i}^{2}-10\mathrm{cos}\left(2\pi {z}_{i}\right)+10\right)+f\_bia{s}_{9}}$ $x\in {\left[-5,5\right]}^{D}$, ${F}_{9}\left({x}^{\ast}\right)=f\_bia{s}_{9}=-330$ |

${F}_{10}$: Shifted Rotated Rastrigin’s Function ${F}_{10}\left(x\right)={\displaystyle \sum _{i=1}^{D}\left({z}_{i}^{2}-10\mathrm{cos}\left(2\pi {z}_{i}\right)+10\right)+f\_bia{s}_{10}}$ $x\in {\left[-5,5\right]}^{D}$. ${F}_{10}\left({x}^{\ast}\right)=f\_bia{s}_{10}=-330$ |

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**Figure 5.**Example cluster implementation for self-adaptive differential evolution algorithm with ensemble of mutation strategies (jEDE) component.

Notation | Function |
---|---|

${F}_{sph}$ | Sphere Function |

${F}_{ros}$ | Rosenbrock’s Function |

${F}_{ack}$ | Ackley’s Function |

${F}_{grw}$ | Griewank’s Function |

${F}_{ras}$ | Rastrigin’s Function |

${F}_{sch}$ | Generalized Schwefel’s Problem 2.26 |

${F}_{sal}$ | Salomon’s Function |

${F}_{wht}$ | Whitely’s Function |

${F}_{pn1}$ | Generalized Penalized Function 1 |

${F}_{pn2}$ | Generalized Penalized Function 2 |

${F}_{1}$ | Shifted Sphere Function |

${F}_{2}$ | Shifted Schwefel’s Problem 1.2 |

${F}_{3}$ | Shifted Rotated High Conditioned Elliptic Function |

${F}_{4}$ | Shifted Schwefel’s Problem 1.2 With Noise in Fitness |

${F}_{5}$ | Schwefel’s Problem 2.6 With Global Optimum on Bounds |

${F}_{6}$ | Shifted Rosenbrock’s Function |

${F}_{7}$ | Shifted Rotated Griewank’s Function without Bounds |

${F}_{8}$ | Shifted Rotated Ackley’s Function with Global Optimum on Bounds |

${F}_{9}$ | Shifted Rastrigin’s Function |

${F}_{10}$ | Shifted Rotated Rastrigin’s Function |

**Table 2.**Comparison of Optimus, Opossum, Silvereye, Galapagos (D = 30, NP = 30, termination: 30 min).

Optimus_jEDE | Opossum_RBFOpt | SilverEye_PSO | Galapagos_GA | Optimal | ||
---|---|---|---|---|---|---|

${F}_{sph}$ | $f\left(x\right)\_min$ | 0.0000000$\times {10}^{0}$ | 1.4000000$\times {10}^{-5}$ | 5.9000000$\times {10}^{-5}$ | 1.1709730$\times {10}^{0}$ | 0 |

$f\left(x\right)\_max$ | 0.0000000$\times {10}^{0}$ | 5.8000000$\times {10}^{-5}$ | 2.7057298$\times {10}^{1}$ | 4.4052130$\times {10}^{0}$ | ||

$f\left(x\right)\_avg$ | 0.0000000$\times {10}^{0}$ | 3.6400000$\times {10}^{-5}$ | 5.4171618$\times {10}^{0}$ | 2.7928586$\times {10}^{0}$ | ||

Std.Dev. | 0.0000000$\times {10}^{0}$ | 1.8039956$\times {10}^{-5}$ | 1.0820072$\times {10}^{1}$ | 1.1492298$\times {10}^{0}$ | ||

FES | 194,520 | 3225 | 31,560 | 34,260 | ||

${F}_{ros}$ | $f\left(x\right)\_min$ | 0.0000000$\times {10}^{0}$ | 2.7485056$\times {10}^{1}$ | 1.6689612$\times {10}^{1}$ | 6.0863438$\times {10}^{3}$ | 0 |

$f\left(x\right)\_max$ | 3.9866240$\times {10}^{0}$ | 2.1030328$\times {10}^{2}$ | 5.8965910$\times {10}^{4}$ | 2.2859534$\times {10}^{4}$ | ||

$f\left(x\right)\_avg$ | 2.3919744$\times {10}^{0}$ | 9.0892522$\times {10}^{1}$ | 1.3859753$\times {10}^{4}$ | 1.3060872$\times {10}^{4}$ | ||

Std.Dev. | 1.9530389$\times {10}^{0}$ | 7.1919037$\times {10}^{1}$ | 2.2886020$\times {10}^{4}$ | 6.7095472$\times {10}^{3}$ | ||

FES | 149,460 | 882 | 26,700 | 35,070 | ||

${F}_{ack}$ | $f\left(x\right)\_min$ | 0.0000000$\times {10}^{0}$ | 3.3550000$\times {10}^{-3}$ | 1.3404210$\times {10}^{0}$ | 5.7470000$\times {10}^{-2}$ | 0 |

$f\left(x\right)\_max$ | 1.3404210$\times {10}^{0}$ | 2.4098540$\times {10}^{0}$ | 3.7340120$\times {10}^{0}$ | 1.0270860$\times {10}^{0}$ | ||

$f\left(x\right)\_avg$ | 2.6808420$\times {10}^{-1}$ | 1.3795174$\times {10}^{0}$ | 2.2482728$\times {10}^{0}$ | 4.8037520$\times {10}^{-1}$ | ||

Std.Dev. | 5.3616840$\times {10}^{-1}$ | 8.5713298$\times {10}^{-1}$ | 9.1850828$\times {10}^{-1}$ | 4.0392221$\times {10}^{-1}$ | ||

FES | 206,370 | 1447 | 38,490 | 28,710 | ||

${F}_{grw}$ | $f\left(x\right)\_min$ | 0.0000000v | 1.5840000$\times {10}^{-3}$ | 3.2081000$\times {10}^{-2}$ | 3.4407200$\times {10}^{-1}$ | 0 |

$f\left(x\right)\_max$ | 0.0000000$\times {10}^{0}$ | 1.7086000$\times {10}^{-2}$ | 2.6292800$\times {10}^{-1}$ | 1.0657060$\times {10}^{0}$ | ||

$f\left(x\right)\_avg$ | 0.0000000$\times {10}^{0}$ | 7.6638000$\times {10}^{-3}$ | 1.2049020$\times {10}^{-1}$ | 8.2474220$\times {10}^{-1}$ | ||

Std.Dev. | 0.0000000$\times {10}^{0}$ | 5.6121253$\times {10}^{-3}$ | 8.1064770$\times {10}^{-2}$ | 2.6131521$\times {10}^{-1}$ | ||

FES | 151,110 | 1089 | 26,610 | 37,410 | ||

${F}_{ras}$ | $f\left(x\right)\_min$ | 4.9747950$\times {10}^{0}$ | 2.5870757$\times {10}^{1}$ | 3.3829188$\times {10}^{1}$ | 7.0535550$\times {10}^{0}$ | 0 |

$f\left(x\right)\_max$ | 2.3879007$\times {10}^{1}$ | 4.1789542$\times {10}^{1}$ | 6.1687356$\times {10}^{1}$ | 2.9072445$\times {10}^{1}$ | ||

$f\left(x\right)\_avg$ | 1.3332448$\times {10}^{1}$ | 3.6218407$\times {10}^{1}$ | 4.8355074$\times {10}^{1}$ | 1.5404780$\times {10}^{1}$ | ||

Std.Dev. | 6.7363920$\times {10}^{0}$ | 5.4349940$\times {10}^{0}$ | 1.1424086$\times {10}^{1}$ | 9.1077975$\times {10}^{0}$ | ||

FES | 206,520 | 4149 | 37,650 | 51,480 | ||

${F}_{sch}$ | $f\left(x\right)\_min$ | 2.3687705$\times {10}^{2}$ | 1.2877414$\times {10}^{3}$ | 4.1089621$\times {10}^{3}$ | 1.9550066$\times {10}^{3}$ | 0 |

$f\left(x\right)\_max$ | 4.7375372$\times {10}^{2}$ | 4.0111803$\times {10}^{3}$ | 6.1589437$\times {10}^{3}$ | 2.7977670$\times {10}^{3}$ | ||

$f\left(x\right)\_avg$ | 4.0269072$\times {10}^{2}$ | 2.7169368$\times {10}^{3}$ | 5.2658793$\times {10}^{3}$ | 2.3201102$\times {10}^{3}$ | ||

Std.Dev. | 9.4750668$\times {10}^{1}$ | 8.8809862$\times {10}^{2}$ | 6.6677783$\times {10}^{2}$ | 2.8681876$\times {10}^{2}$ | ||

FES | 148,140 | 1487 | 27,210 | 35,940 | ||

${F}_{sal}$ | $f\left(x\right)\_min$ | 1.9987300$\times {10}^{-1}$ | 2.8070190$\times {10}^{0}$ | 2.9987300$\times {10}^{-1}$ | 1.4998750$\times {10}^{0}$ | 0 |

$f\left(x\right)\_max$ | 4.9987300$\times {10}^{-1}$ | 4.3000810$\times {10}^{0}$ | 4.9987300$\times {10}^{-1}$ | 2.8375760$\times {10}^{0}$ | ||

$f\left(x\right)\_avg$ | 3.1987300$\times {10}^{-1}$ | 3.4413810$\times {10}^{0}$ | 3.7987300$\times {10}^{-1}$ | 2.0682340$\times {10}^{0}$ | ||

Std.Dev. | 1.1661904$\times {10}^{-1}$ | 5.6623101$\times {10}^{-1}$ | 7.4833148$\times {10}^{-2}$ | 6.1557512$\times {10}^{-1}$ | ||

FES | 201,720 | 4769 | 38,640 | 51,360 | ||

${F}_{wht}$ | $f\left(x\right)\_min$ | 2.3704633$\times {10}^{1}$ | 9.6592754$\times {10}^{2}$ | 1.8455490$\times {10}^{2}$ | 2.3632742$\times {10}^{5}$ | 0 |

$f\left(x\right)\_max$ | 2.4040716$\times {10}^{2}$ | 1.6904059$\times {10}^{3}$ | 6.2776811$\times {10}^{2}$ | 5.5055000$\times {10}^{8}$ | ||

$f\left(x\right)\_avg$ | 1.0789137$\times {10}^{2}$ | 1.2610498$\times {10}^{3}$ | 4.0698894$\times {10}^{2}$ | 2.7440867$\times {10}^{8}$ | ||

Std.Dev. | 7.4951993$\times {10}^{1}$ | 2.6984398$\times {10}^{2}$ | 1.6923390$\times {10}^{2}$ | 2.2575944$\times {10}^{8}$ | ||

FES | 146,640 | 728 | 23,250 | 29,730 | ||

${F}_{pn1}$ | $f\left(x\right)\_min$ | 0.0000000$\times {10}^{0}$ | 2.9057000$\times {10}^{-2}$ | 3.1283800$\times {10}^{-1}$ | 1.4510000$\times {10}^{-3}$ | 0 |

$f\left(x\right)\_max$ | 0.0000000$\times {10}^{0}$ | 9.0392970$\times {10}^{0}$ | 1.3487570$\times {10}^{0}$ | 1.7632000$\times {10}^{-2}$ | ||

$f\left(x\right)\_avg$ | 0.0000000$\times {10}^{0}$ | 2.8243854$\times {10}^{0}$ | 6.7680180$\times {10}^{-1}$ | 6.0638000$\times {10}^{-3}$ | ||

Std.Dev. | 0.0000000$\times {10}^{0}$ | 3.1774566$\times {10}^{0}$ | 4.2868737$\times {10}^{-1}$ | 6.0403379$\times {10}^{-3}$ | ||

FES | 203,880 | 1394 | 39,420 | 57,720 | ||

${F}_{pn2}$ | $f\left(x\right)\_min$ | 0.0000000$\times {10}^{0}$ | 2.0400434$\times {10}^{1}$ | 1.0000000$\times {10}^{-11}$ | 1.8037300$\times {10}^{-1}$ | 0 |

$f\left(x\right)\_max$ | 1.0987000$\times {10}^{-2}$ | 2.8693232$\times {10}^{1}$ | 9.3079800$\times {10}^{-1}$ | 2.7208440$\times {10}^{0}$ | ||

$f\left(x\right)\_avg$ | 2.1974000$\times {10}^{-3}$ | 2.5384324$\times {10}^{1}$ | 2.2552480$\times {10}^{-1}$ | 1.0041520$\times {10}^{0}$ | ||

Std.Dev. | 4.3948000$\times {10}^{-3}$ | 3.4851206$\times {10}^{0}$ | 3.5679494$\times {10}^{-1}$ | 9.4298611$\times {10}^{-1}$ | ||

FES | 148,380 | 639 | 29,040 | 41,520 | ||

${F}_{1}$ | $f\left(x\right)\_min$ | −4.5000000$\times {10}^{2}$ | −4.4999898$\times {10}^{2}$ | 2.6232595$\times {10}^{2}$ | −4.4995998$\times {10}^{2}$ | −450 |

$f\left(x\right)\_max$ | −4.5000000$\times {10}^{2}$ | −4.4999478$\times {10}^{2}$ | 8.0377273$\times {10}^{3}$ | −4.4988406$\times {10}^{2}$ | ||

$f\left(x\right)\_avg$ | −4.5000000$\times {10}^{2}$ | −4.4999680$\times {10}^{2}$ | 4.0824562$\times {10}^{3}$ | −4.4992015$\times {10}^{2}$ | ||

Std.Dev. | 0.0000000$\times {10}^{0}$ | 1.4829922$\times {10}^{-3}$ | 2.9460423$\times {10}^{3}$ | 3.0428108$\times {10}^{-2}$ | ||

FES | 198,060 | 6156 | 45,580 | 66,180 | ||

${F}_{2}$ | $f\left(x\right)\_min$ | −4.5000000$\times {10}^{2}$ | 3.4590652$\times {10}^{4}$ | −3.8035693$\times {10}^{2}$ | 6.8476195$\times {10}^{3}$ | −450 |

$f\left(x\right)\_max$ | −4.5000000$\times {10}^{2}$ | 4.7978174$\times {10}^{4}$ | 5.2590674$\times {10}^{2}$ | 1.2302281$\times {10}^{4}$ | ||

$f\left(x\right)\_avg$ | −4.5000000$\times {10}^{2}$ | 4.3226072$\times {10}^{4}$ | −1.2838464$\times {10}^{2}$ | 1.0174618$\times {10}^{4}$ | ||

Std.Dev. | 0.0000000$\times {10}^{0}$ | 4.9030645$\times {10}^{3}$ | 3.3183646$\times {10}^{2}$ | 1.8557926$\times {10}^{3}$ | ||

FES | 146,010 | 1061 | 33,840 | 50,160 | ||

${F}_{3}$ | $f\left(x\right)\_min$ | 6.0045376$\times {10}^{4}$ | 1.5561000$\times {10}^{7}$ | 1.9264000$\times {10}^{6}$ | 1.1250000$\times {10}^{7}$ | −450 |

$f\left(x\right)\_max$ | 2.4850013$\times {10}^{5}$ | 7.5084000$\times {10}^{7}$ | 8.0820000$\times {10}^{6}$ | 2.7772000$\times {10}^{7}$ | ||

$f\left(x\right)\_avg$ | 1.2857393$\times {10}^{5}$ | 3.7380600$\times {10}^{7}$ | 4.5525000$\times {10}^{6}$ | 1.7212200$\times {10}^{7}$ | ||

Std.Dev. | 6.4175884$\times {10}^{4}$ | 2.0647812$\times {10}^{7}$ | 2.0206526$\times {10}^{6}$ | 5.6281541$\times {10}^{6}$ | ||

FES | 205,260 | 1293 | 48,030 | 66,000 | ||

${F}_{4}$ | $f\left(x\right)\_min$ | −4.5000000$\times {10}^{2}$ | 2.8373782$\times {10}^{4}$ | −4.2715712$\times {10}^{2}$ | 7.8877685$\times {10}^{3}$ | −450 |

$f\left(x\right)\_max$ | −4.5000000$\times {10}^{2}$ | 3.9404224$\times {10}^{4}$ | 4.0484178$\times {10}^{3}$ | 1.1191542$\times {10}^{4}$ | ||

$f\left(x\right)\_avg$ | −4.5000000$\times {10}^{2}$ | 3.2359668$\times {10}^{4}$ | 5.1724092$\times {10}^{2}$ | 9.4535270$\times {10}^{3}$ | ||

Std.Dev. | 0.0000000$\times {10}^{0}$ | 4.0412734$\times {10}^{3}$ | 1.7663033$\times {10}^{3}$ | 1.2966977$\times {10}^{3}$ | ||

FES | 147,240 | 1055 | 35,610 | 53,520 | ||

${F}_{5}$ | $f\left(x\right)\_min$ | 9.3362001$\times {10}^{2}$ | 4.7527012$\times {10}^{3}$ | 7.5856684$\times {10}^{3}$ | 1.8506721$\times {10}^{4}$ | -310 |

$f\left(x\right)\_max$ | 2.5603668$\times {10}^{3}$ | 5.8813877$\times {10}^{3}$ | 1.2910221$\times {10}^{4}$ | 2.4057172$\times {10}^{4}$ | ||

$f\left(x\right)\_avg$ | 2.0333032$\times {10}^{3}$ | 5.3824799$\times {10}^{3}$ | 9.4390617$\times {10}^{3}$ | 2.0151105$\times {10}^{4}$ | ||

Std.Dev. | 570.1512256 | 438.2070353 | 2031.289127 | 2004.100331 | ||

FES | 195,720 | 1891 | 47,550 | 67,560 | ||

${F}_{6}$ | $f\left(x\right)\_min$ | 3.9000000$\times {10}^{2}$ | 1.4168473$\times {10}^{3}$ | 5.0073093$\times {10}^{2}$ | 9.7856127$\times {10}^{2}$ | 390 |

$f\left(x\right)\_max$ | 3.9398662$\times {10}^{2}$ | 1.3904779$\times {10}^{4}$ | 4.1540000$\times {10}^{9}$ | 9.6995775$\times {10}^{3}$ | ||

$f\left(x\right)\_avg$ | 3.9159465$\times {10}^{2}$ | 8.7212119$\times {10}^{3}$ | 9.8402870$\times {10}^{8}$ | 5.6395846$\times {10}^{3}$ | ||

Std.Dev. | 1.9530389$\times {10}^{0}$ | 4.6035484$\times {10}^{3}$ | 1.5972516$\times {10}^{8}$ | 3.5378379$\times {10}^{3}$ | ||

FES | 148,260 | 687 | 33,540 | 48,810 | ||

${F}_{7}$ | $f\left(x\right)\_min$ | 4.5162886$\times {10}^{3}$ | 4.5162896$\times {10}^{3}$ | 5.8669417$\times {10}^{3}$ | 4.5172240$\times {10}^{3}$ | -180 |

$f\left(x\right)\_max$ | 4.5162886$\times {10}^{3}$ | 4.5162985$\times {10}^{3}$ | 7.2432580$\times {10}^{3}$ | 4.5290168$\times {10}^{3}$ | ||

$f\left(x\right)\_avg$ | 4.5162886$\times {10}^{3}$ | 4.5162936$\times {10}^{3}$ | 6.5251090$\times {10}^{3}$ | 4.5222540$\times {10}^{3}$ | ||

Std.Dev. | 0.0000000$\times {10}^{0}$ | 3.1911420$\times {10}^{-3}$ | 5.4380701$\times {10}^{2}$ | 4.7496031$\times {10}^{0}$ | ||

FES | 200,820 | 10,108 | 42,060 | 58,290 | ||

${F}_{8}$ | $f\left(x\right)\_min$ | −1.1905178$\times {10}^{2}$ | −1.1910166$\times {10}^{2}$ | −1.1901775$\times {10}^{2}$ | −1.1940297$\times {10}^{2}$ | -140 |

$f\left(x\right)\_max$ | −1.1902135$\times {10}^{2}$ | −1.1876717$\times {10}^{2}$ | −1.1892500$\times {10}^{2}$ | −1.1906700$\times {10}^{2}$ | ||

$f\left(x\right)\_avg$ | −1.1903319$\times {10}^{2}$ | −1.1889866$\times {10}^{2}$ | −1.1899553$\times {10}^{2}$ | −1.1919711$\times {10}^{2}$ | ||

Std.Dev. | 1.0538581$\times {10}^{-2}$ | 1.1070562$\times {10}^{-1}$ | 3.5483336$\times {10}^{-2}$ | 1.4127484$\times {10}^{-2}$ | ||

FES | 149,670 | 2018 | 35,580 | 52,020 | ||

${F}_{9}$ | $f\left(x\right)\_min$ | −3.1706554$\times {10}^{2}$ | −2.5827181$\times {10}^{2}$ | −2.3690804$\times {10}^{2}$ | −3.1677970$\times {10}^{2}$ | -330 |

$f\left(x\right)\_max$ | −3.1209075$\times {10}^{2}$ | −2.3567358$\times {10}^{2}$ | −1.6682751$\times {10}^{2}$ | −3.1164785$\times {10}^{2}$ | ||

$f\left(x\right)\_avg$ | −3.1527462$\times {10}^{2}$ | −2.4625749$\times {10}^{2}$ | −1.8917798$\times {10}^{2}$ | −3.1499413$\times {10}^{2}$ | ||

Std.Dev. | 1.7117897$\times {10}^{0}$ | 8.0334497$\times {10}^{0}$ | 2.5285354$\times {10}^{1}$ | 1.8617799$\times {10}^{0}$ | ||

FES | 212,160 | 5577 | 47,610 | 67,560 | ||

${F}_{10}$ | $f\left(x\right)\_min$ | −2.7030257$\times {10}^{2}$ | −2.3528777$\times {10}^{2}$ | −1.3299956$\times {10}^{2}$ | 4.0215414$\times {10}^{1}$ | -330 |

$f\left(x\right)\_max$ | −2.2751946$\times {10}^{2}$ | −1.3298172$\times {10}^{2}$ | −8.1262211$\times {10}^{1}$ | 1.8543670$\times {10}^{2}$ | ||

$f\left(x\right)\_avg$ | −2.5139841$\times {10}^{2}$ | −1.8578676$\times {10}^{2}$ | −1.0572303$\times {10}^{2}$ | 1.1181316$\times {10}^{2}$ | ||

Std.Dev. | 1.4307998$\times {10}^{1}$ | 3.9394042$\times {10}^{1}$ | 1.8528544$\times {10}^{1}$ | 5.7458318$\times {10}^{1}$ | ||

FES | 146,820 | 1192 | 35,220 | 53,070 |

Notation | Design Parameter | Min | Max | Type |
---|---|---|---|---|

x1–x13 | Coordinates of control points in axis 1 | −2.00 | 2.00 | Continues |

x14–x26 | Coordinates of control points in axis 2 | −2.00 | 2.00 | Continues |

x27–x39 | Coordinates of control points in axis 3 | −2.00 | 2.00 | Continues |

x40–x52 | Coordinates of control points in axis 4 | −2.00 | 2.00 | Continues |

x53–x65 | Coordinates of control points in axis 5 | −2.00 | 2.00 | Continues |

x66 | Division amount on u direction | 3 | 10 | Discrete |

x67 | Division amount on v direction | 3 | 10 | Discrete |

x68 | Depth of the truss | 0.50 | 1.00 | Continues |

x69 | Height of the cross-section | 10.00 | 30.00 | Continues |

x70 | Upper and lower width of the cross-section | 10.00 | 30.00 | Continues |

**Table 4.**Comparison of Optimus, Opossum, Silvereye, Galapagos (D = 70, NP = 50, termination: 30 min).

Optimus jEDE | Opossum RBFOpt | Silve Eye_PSO | Galapagos GA | ||
---|---|---|---|---|---|

Design problem | $f\left(x\right)\_min$ | 6.21637$\times {10}^{3}$ | 2.25410$\times {10}^{4}$ | 6.89062$\times {10}^{5}$ | 6.74560$\times {10}^{5}$ |

$g\left(x\right)$ | 0.0994 | 0.0996 | 0.6674 | 0.9273 | |

FES | 31,800 | 5102 | 17,100 | 19,000 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cubukcuoglu, C.; Ekici, B.; Tasgetiren, M.F.; Sariyildiz, S.
OPTIMUS: Self-Adaptive Differential Evolution with Ensemble of Mutation Strategies for Grasshopper Algorithmic Modeling. *Algorithms* **2019**, *12*, 141.
https://doi.org/10.3390/a12070141

**AMA Style**

Cubukcuoglu C, Ekici B, Tasgetiren MF, Sariyildiz S.
OPTIMUS: Self-Adaptive Differential Evolution with Ensemble of Mutation Strategies for Grasshopper Algorithmic Modeling. *Algorithms*. 2019; 12(7):141.
https://doi.org/10.3390/a12070141

**Chicago/Turabian Style**

Cubukcuoglu, Cemre, Berk Ekici, Mehmet Fatih Tasgetiren, and Sevil Sariyildiz.
2019. "OPTIMUS: Self-Adaptive Differential Evolution with Ensemble of Mutation Strategies for Grasshopper Algorithmic Modeling" *Algorithms* 12, no. 7: 141.
https://doi.org/10.3390/a12070141