# An Empirical Investigation on Evolutionary Algorithm Evolving Developmental Timings

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. The Evolutionary Algorithm Evolving Developmental Timings

#### 3.1. Individual

#### 3.2. Individual Development

#### 3.3. Crossover Operator

#### 3.4. Mutation and Selection Operators

## 4. Simulations for Performance Evaluation

#### 4.1. Test Problems

#### 4.1.1. Hierarchical If-And-Only-If Function (HIFF)

#### 4.1.2. Hierarchical Trap Function (HTRAP)

#### 4.1.3. Hierarchically Dependent Function (HDEP)

- 1.
- Generate a connected graph with some topology. The number of nodes in the graph is equal to the number of variables, that is, the length of a binary string as a solution candidate, ℓ.
- 2.
- Assign position numbers of a binary string as a solution candidate, $1,2,\dots ,\ell $, to each of the ℓ nodes on the generated graph.
- 3.
- Define Lm-bit problems with a factor of deception on the generated graph. In the graph, nodes of higher degree form higher hierarchies. Degree means the number of edges. If multiple nodes of the same degree are included in an m-bit problem, the nodes with smaller position numbers become those of higher degree.

- Assign bits of a solution candidate to corresponding nodes in the graph of the problem instance.
- Calculate the sum of fitness values of Lm-bit problems defined on the graph and then divide the sum by L. A solution candidate of an m-bit problem consists of m bits that the m nodes sequentially connected by edges have. More concretely, the solution candidate is formed by arranging m bits that the m nodes have from higher to lower in terms of hierarchy.
- Set the value obtained by the division to be a fitness value of a solution candidate of the entire problem instance.

#### 4.1.4. N-K Landscape Function (K = 4) (NKL-K4)

#### 4.1.5. Multidimensional Knapsack Problem (MKP)

#### 4.2. Algorithm for Comparison

#### 4.3. Configurations

#### 4.4. Results

#### 4.5. Discussion

#### 4.5.1. Why the EDT Solves MKP Better Than the LTGA

#### 4.5.2. Why the EDT Solves HIFF and HTRAP Worse Than the LTGA

#### 4.5.3. Why the Performances of the EDT and the CBGA Are Similar

#### 4.5.4. What Configurations Are Better for the EDT

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**An application example of the crossover operator of the evolutionary algorithm evolving developmental timing (EDT).

**Figure 5.**The degree distributions of three graphs used in the HDEP. The numbers of nodes in the three graphs, which are equivalent to the problem size, ℓ, are 20, 30, and 40. They are all power-law distribution.

**Figure 6.**The fitness value of g in the sequential five bits, (${g}_{r1}$, ${g}_{r2}$, $g$, ${g}_{\ell 1}$, ${g}_{\ell 2}$), in NKL-K4.

**Figure 7.**The convergence of the phenotypic values when the problem was successfully solved, and fitness convergence curves for the runs.

**Figure 8.**Examples of the time variation of a dominant of each variable, 0 or 1, in the development of an individual at the first and the last generations. The black bold line in each figure represents the average of the ratios of all variables.

**Figure 9.**The convergence curve of phenotypic values corresponding to the nodes with the three larger degrees when solving HDEP.

**Table 1.**The fitness of g in the sequential five bits, (${g}_{r1}$, ${g}_{r2}$, $g$, ${g}_{\ell 1}$, ${g}_{\ell 2}$), in NKL-K4. The five-bit string in the table represents ($g$, ${g}_{r1}$, ${g}_{r2}$, ${g}_{\ell 1}$, ${g}_{\ell 2}$).

5-Bit | Fitness | 5-Bit | Fitness | 5-Bit | Fitness | 5-Bit | Fitness |
---|---|---|---|---|---|---|---|

00000 | 0.036486 | 01000 | 0.258027 | 10000 | 0.315626 | 11000 | 0.101828 |

00001 | 0.833081 | 01001 | 0.467604 | 10001 | 0.575035 | 11001 | 0.533017 |

00010 | 0.267900 | 01010 | 0.243886 | 10010 | 0.704985 | 11010 | 0.118997 |

00011 | 0.011235 | 01011 | 0.040266 | 10011 | 0.283613 | 11011 | 0.546785 |

00100 | 0.882766 | 01100 | 0.178573 | 10100 | 0.661520 | 11100 | 0.516638 |

00101 | 0.213545 | 01101 | 0.803215 | 10101 | 0.175868 | 11101 | 0.707389 |

00110 | 0.778439 | 01110 | 0.903812 | 10110 | 0.979191 | 11110 | 0.038014 |

00111 | 0.537816 | 01111 | 0.262323 | 10111 | 0.886160 | 11111 | 0.452097 |

EDT | SGA | CBGA | ecGA | LTGA | |||||
---|---|---|---|---|---|---|---|---|---|

HIFF (ℓ = 32) | |||||||||

P = 4 | P = 60 | P = 80 | P = 100 | P = 2000 | P = 2000 | P = 5000 | P = 40 | ||

${T}_{c}$=$\lceil \ell /2\rceil $ | ${N}_{r}$ | 4 | 18 | 23 | 20 | 9 | 30 | 30 | 30 |

${N}_{e}$ | 5.8$\times {10}^{3}$ | 4.3$\times {10}^{3}$ | 5.3$\times {10}^{3}$ | 6.3$\times {10}^{3}$ | 1.3$\times {10}^{5}$ | 1.5$\times {10}^{5}$ | 5.4$\times {10}^{4}$ | 2.0$\times {10}^{3}$ | |

$S{D}_{e}$ | 2.5$\times {10}^{3}$ | 0.6$\times {10}^{4}$ | 1.1$\times {10}^{4}$ | 1.2$\times {10}^{4}$ | 3.0$\times {10}^{3}$ | $1.2\times {10}^{5}$ | 0.7$\times {10}^{4}$ | - | |

${F}_{a}$ | 1.5$\times {10}^{2}$ | 1.8$\times {10}^{2}$ | 1.8$\times {10}^{2}$ | 1.8$\times {10}^{2}$ | 164.2 | 192 | 192 | 192 | |

$S{D}_{f}$ | 0.2$\times {10}^{2}$ | 0.2$\times {10}^{2}$ | 0.2$\times {10}^{2}$ | 0.2$\times {10}^{2}$ | 19.3 | 0 | 0 | 0 | |

${T}_{c}$=$\lceil \ell /4\rceil $ | ${N}_{r}$ | 2 | 11 | 17 | 20 | ||||

${N}_{e}$ | 6.2$\times {10}^{3}$ | 3.4$\times {10}^{4}$ | 4.6$\times {10}^{4}$ | 5.6$\times {10}^{4}$ | |||||

$S{D}_{e}$ | 0.9$\times {10}^{2}$ | 0.5$\times {10}^{4}$ | 0.7$\times {10}^{4}$ | 0.9$\times {10}^{4}$ | |||||

${F}_{a}$ | 1.5$\times {10}^{2}$ | 1.7$\times {10}^{2}$ | 1.8$\times {10}^{2}$ | 1.8$\times {10}^{2}$ | |||||

$S{D}_{f}$ | 0.1$\times {10}^{2}$ | 0.2$\times {10}^{2}$ | 0.2$\times {10}^{2}$ | 0.2$\times {10}^{2}$ | |||||

${T}_{c}$=$\lceil \ell /8\rceil $ | ${N}_{r}$ | 14 | 10 | 12 | 16 | ||||

${N}_{e}$ | 5.2$\times {10}^{5}$ | 1.7$\times {10}^{5}$ | 2.6$\times {10}^{5}$ | 3.3$\times {10}^{5}$ | |||||

$S{D}_{e}$ | 3.2$\times {10}^{5}$ | 0.6$\times {10}^{5}$ | 1.3$\times {10}^{5}$ | 1.4$\times {10}^{5}$ | |||||

${F}_{a}$ | 1.7$\times {10}^{2}$ | 1.7$\times {10}^{2}$ | 1.7$\times {10}^{2}$ | 1.8$\times {10}^{2}$ | |||||

$S{D}_{f}$ | 0.2$\times {10}^{2}$ | 0.2$\times {10}^{2}$ | 0.2$\times {10}^{2}$ | 0.2$\times {10}^{2}$ | |||||

HIFF (ℓ = 64) | |||||||||

P = 4 | P = 60 | P = 80 | P = 100 | P = 2000 | P = 2000 | P = 5000 | P = 50 | ||

${T}_{c}$=$\lceil \ell /2\rceil $ | ${N}_{r}$ | 0 | 5 | 8 | 8 | 3 | 15 | 1 | 30 |

${N}_{e}$ | - | 2.8$\times {10}^{5}$ | 3.8$\times {10}^{5}$ | 4.2$\times {10}^{5}$ | 1.5$\times {10}^{6}$ | 1.5$\times {10}^{6}$ | 1.3$\times {10}^{5}$ | 9.0$\times {10}^{3}$ | |

$S{D}_{e}$ | - | 0.3$\times {10}^{5}$ | 0.8$\times {10}^{5}$ | 0.8$\times {10}^{5}$ | 9.6$\times {10}^{5}$ | 1.1$\times {10}^{6}$ | 0.0 | - | |

${F}_{a}$ | 3.2$\times {10}^{2}$ | 3.6$\times {10}^{2}$ | 3.8$\times {10}^{2}$ | 3.8$\times {10}^{2}$ | 332.8 | 416.0 | 338.4 | 448 | |

$S{D}_{f}$ | 0.2$\times {10}^{2}$ | 0.2$\times {10}^{2}$ | 0.2$\times {10}^{2}$ | 0.2$\times {10}^{2}$ | 49.6 | 32.0 | 29.3 | 0 | |

${T}_{c}$=$\lceil \ell /4\rceil $ | ${N}_{r}$ | 1 | 6 | 7 | 5 | ||||

${N}_{e}$ | 8.9$\times {10}^{5}$ | 4.2$\times {10}^{5}$ | 2.3$\times {10}^{5}$ | 2.5$\times {10}^{5}$ | |||||

$S{D}_{e}$ | - | 5.8$\times {10}^{5}$ | 0.9$\times {10}^{5}$ | 0.4$\times {10}^{5}$ | |||||

${F}_{a}$ | 3.2$\times {10}^{2}$ | 3.6$\times {10}^{2}$ | 3.6$\times {10}^{2}$ | 3.6$\times {10}^{2}$ | |||||

$S{D}_{f}$ | 0.4$\times {10}^{2}$ | 0.5$\times {10}^{2}$ | 0.5$\times {10}^{2}$ | 0.4$\times {10}^{2}$ | |||||

${T}_{c}$=$\lceil \ell /8\rceil $ | ${N}_{r}$ | 0 | 0 | 3 | 2 | ||||

${N}_{e}$ | - | - | 3.2$\times {10}^{5}$ | 3.5$\times {10}^{2}$ | |||||

$S{D}_{e}$ | - | - | 0.4$\times {10}^{5}$ | 0.2$\times {10}^{5}$ | |||||

${F}_{a}$ | 3.2$\times {10}^{2}$ | 3.4$\times {10}^{2}$ | 3.5$\times {10}^{2}$ | 3.5$\times {10}^{2}$ | |||||

$S{D}_{f}$ | 0.3$\times {10}^{2}$ | 0.3$\times {10}^{2}$ | 0.4$\times {10}^{2}$ | 0.3$\times {10}^{2}$ | |||||

HIFF (ℓ = 128) | |||||||||

P = 4 | P = 60 | P = 80 | P = 100 | P = 2000 | P = 2000 | P = 5000 | P = 60 | ||

${T}_{c}$=$\lceil \ell /2\rceil $ | ${N}_{r}$ | 0 | 0 | 0 | 0 | 0 | 0 | - | 30 |

${N}_{e}$ | - | - | - | - | - | - | - | 2.5$\times {10}^{4}$ | |

$S{D}_{e}$ | - | - | - | - | - | - | - | - | |

${F}_{a}$ | 5.8$\times {10}^{2}$ | 6.8$\times {10}^{2}$ | 7.0$\times {10}^{2}$ | 6.7$\times {10}^{2}$ | 471.8 | 706.0 | - | 1024 | |

$S{D}_{f}$ | 0.4$\times {10}^{2}$ | 0.5$\times {10}^{2}$ | 0.5$\times {10}^{2}$ | 0.5$\times {10}^{2}$ | 23.4 | 71.2 | - | 0 | |

${T}_{c}$=$\lceil \ell /4\rceil $ | ${N}_{r}$ | 0 | 0 | 0 | 0 | ||||

${N}_{e}$ | - | - | - | - | |||||

$S{D}_{e}$ | - | - | - | - | |||||

${F}_{a}$ | 6.2$\times {10}^{2}$ | 7.3$\times {10}^{2}$ | 7.3$\times {10}^{2}$ | 7.4$\times {10}^{2}$ | |||||

$S{D}_{f}$ | 0.5$\times {10}^{2}$ | 0.8$\times {10}^{2}$ | 0.5$\times {10}^{2}$ | 0.7$\times {10}^{2}$ | |||||

${T}_{c}$=$\lceil \ell /8\rceil $ | ${N}_{r}$ | 0 | 0 | 0 | 0 | ||||

${N}_{e}$ | - | - | - | - | |||||

$S{D}_{e}$ | - | - | - | - | |||||

${F}_{a}$ | 5.8$\times {10}^{2}$ | 6.7$\times {10}^{2}$ | 6.9$\times {10}^{2}$ | 7.0$\times {10}^{2}$ | |||||

$S{D}_{f}$ | 0.5$\times {10}^{2}$ | 0.6$\times {10}^{2}$ | 0.5$\times {10}^{2}$ | 0.6$\times {10}^{2}$ |

EDT | SGA | CBGA | ecGA | LTGA | |||||
---|---|---|---|---|---|---|---|---|---|

HTRAP (ℓ = 9) | |||||||||

P = 4 | P = 60 | P = 80 | P = 100 | P = 2000 | P = 2000 | P = 5000 | - | ||

${T}_{c}$=$\lceil \ell /2\rceil $ | ${N}_{r}$ | 30 | 30 | 30 | 30 | 30 | 30 | 30 | - |

${N}_{e}$ | 1.2$\times {10}^{5}$ | 4.9$\times {10}^{2}$ | 4.5$\times {10}^{2}$ | 6.0$\times {10}^{2}$ | 3.3$\times {10}^{2}$ | 2.0$\times {10}^{3}$ | 4.0$\times {10}^{2}$ | - | |

$S{D}_{e}$ | 1.8$\times {10}^{5}$ | 4.2$\times {10}^{2}$ | 3.3$\times {10}^{2}$ | 6.3$\times {10}^{2}$ | 3.6$\times {10}^{2}$ | 1.7$\times {10}^{1}$ | 4.0$\times {10}^{2}$ | - | |

${F}_{a}$ | 6 | 6 | 6 | 6 | 6 | 6 | 6 | - | |

$S{D}_{f}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | - | |

${T}_{c}$=$\lceil \ell /4\rceil $ | ${N}_{r}$ | 30 | 30 | 30 | 30 | ||||

${N}_{e}$ | 2.3$\times {10}^{3}$ | 5.5$\times {10}^{2}$ | 4.9$\times {10}^{2}$ | 4.1$\times {10}^{2}$ | |||||

$S{D}_{e}$ | 5.2$\times {10}^{3}$ | 4.4$\times {10}^{2}$ | 4.5$\times {10}^{2}$ | 3.0$\times {10}^{2}$ | |||||

${F}_{a}$ | 6 | 6 | 6 | 6 | |||||

$S{D}_{f}$ | 0 | 0 | 0 | 0 | |||||

${T}_{c}$=$\lceil \ell /8\rceil $ | ${N}_{r}$ | 30 | 30 | 30 | 30 | ||||

${N}_{e}$ | 6.9$\times {10}^{2}$ | 6.7$\times {10}^{2}$ | 5.1$\times {10}^{2}$ | 4.8$\times {10}^{2}$ | |||||

$S{D}_{e}$ | 6.3$\times {10}^{2}$ | 6.2$\times {10}^{2}$ | 4.5$\times {10}^{2}$ | 4.0$\times {10}^{2}$ | |||||

${F}_{a}$ | 6 | 6 | 6 | 6 | |||||

$S{D}_{f}$ | 0 | 0 | 0 | 0 | |||||

HTRAP (ℓ = 27) | |||||||||

P = 4 | P = 60 | P = 80 | P = 100 | P = 2000 | P = 2000 | P = 5000 | P = 100 | ||

${T}_{c}$=$\lceil \ell /2\rceil $ | ${N}_{r}$ | 0 | 3 | 3 | 7 | 0 | 30 | 27 | 30 |

${N}_{e}$ | - | 3.3$\times {10}^{4}$ | 4.3$\times {10}^{4}$ | 4.3$\times {10}^{4}$ | - | 2.4$\times {10}^{6}$ | 3.7$\times {10}^{4}$ | 1.0$\times {10}^{4}$ | |

$S{D}_{e}$ | - | 0.9$\times {10}^{4}$ | 0.2$\times {10}^{4}$ | 1.0$\times {10}^{4}$ | - | 4.2$\times {10}^{5}$ | 0.9$\times {10}^{4}$ | - | |

${F}_{a}$ | 2.3$\times 10$ | 2.5$\times 10$ | 2.6$\times 10$ | 2.6$\times 10$ | 26.1 | 27 | 26.9 | 27 | |

$S{D}_{f}$ | 3.0 | 1.7 | 1.7 | 1.4 | 0.0 | 0 | 0.3 | 0 | |

${T}_{c}$=$\lceil \ell /4\rceil $ | ${N}_{r}$ | 0 | 2 | 1 | 3 | ||||

${N}_{e}$ | - | 2.7$\times {10}^{4}$ | 2.7$\times {10}^{4}$ | 4.3$\times {10}^{4}$ | |||||

$S{D}_{e}$ | - | 0.2$\times {10}^{4}$ | - | 0.4$\times {10}^{4}$ | |||||

${F}_{a}$ | 2.3$\times 10$ | 2.5$\times 10$ | 2.6$\times 10$ | 2.5$\times 10$ | |||||

$S{D}_{f}$ | 3.1 | 2.2 | 1.2 | 1.5 | |||||

${T}_{c}$=$\lceil \ell /8\rceil $ | ${N}_{r}$ | 0 | 1 | 1 | 0 | ||||

${N}_{e}$ | - | 1.1$\times {10}^{5}$ | 1.7$\times {10}^{5}$ | - | |||||

$S{D}_{e}$ | - | - | - | - | |||||

${F}_{a}$ | 2.3$\times 10$ | 2.5$\times 10$ | 2.5$\times 10$ | 2.5$\times 10$ | |||||

$S{D}_{f}$ | 2.6 | 1.8 | 1.4 | 1.7 | |||||

HTRAP (ℓ = 81) | |||||||||

P = 4 | P = 60 | P = 80 | P = 100 | P = 2000 | P = 2000 | P = 5000 | P = 110 | ||

${T}_{c}$=$\lceil \ell /2\rceil $ | ${N}_{r}$ | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 30 |

${N}_{e}$ | - | - | - | 1.0$\times {10}^{6}$ | - | - | - | 6.0$\times {10}^{4}$ | |

$S{D}_{e}$ | - | - | - | - | - | - | - | - | |

${F}_{a}$ | 7.3$\times 10$ | 7.7$\times 10$ | 8.3$\times 10$ | 8.5$\times 10$ | 105.3 | 105.3 | 105.3 | 108 | |

$S{D}_{f}$ | 7.3 | 1.0$\times 10$ | 1.2$\times 10$ | 1.5$\times 10$ | 0.0 | 0.0 | 0.0 | 0 | |

${T}_{c}$=$\lceil \ell /4\rceil $ | ${N}_{r}$ | 0 | 0 | 0 | 0 | ||||

${N}_{e}$ | - | - | - | - | |||||

$S{D}_{e}$ | - | - | - | - | |||||

${F}_{a}$ | 7.4$\times 10$ | 7.9$\times 10$ | 8.1$\times 10$ | 7.8$\times 10$ | |||||

$S{D}_{f}$ | 7.2 | 1.3$\times 10$ | 1.2$\times 10$ | 9.7 | |||||

${T}_{c}$=$\lceil \ell /8\rceil $ | ${N}_{r}$ | 0 | 0 | 0 | 0 | ||||

${N}_{e}$ | - | - | - | - | |||||

$S{D}_{e}$ | - | - | - | - | |||||

${F}_{a}$ | 6.2$\times 10$ | 7.7$\times 10$ | 7.7$\times 10$ | 8.0$\times 10$ | |||||

$S{D}_{f}$ | 6.3 | 1.0$\times 10$ | 1.2$\times 10$ | 1.2$\times 10$ |

EDT | SGA | CBGA | ecGA | LTGA | |||||
---|---|---|---|---|---|---|---|---|---|

HDEP (ℓ = 20) | |||||||||

P = 4 | P = 60 | P = 80 | P = 100 | P = 2000 | P = 2000 | P = 5000 | P = 200 | ||

${T}_{c}$=$\lceil \ell /2\rceil $ | ${N}_{r}$ | 27 | 30 | 30 | 30 | 8 | 30 | 17 | 30 |

${N}_{e}$ | 5.4$\times {10}^{5}$ | 3.5$\times {10}^{4}$ | 2.1$\times {10}^{4}$ | 3.2$\times {10}^{4}$ | 4.7$\times {10}^{4}$ | 8.4$\times {10}^{4}$ | 2.8$\times {10}^{4}$ | 3.8$\times {10}^{3}$ | |

$S{D}_{e}$ | 8.6$\times {10}^{5}$ | 9.5$\times {10}^{4}$ | 0.7$\times {10}^{4}$ | 3.3$\times {10}^{4}$ | 2.8$\times {10}^{4}$ | 5.0$\times {10}^{4}$ | 0.8$\times {10}^{4}$ | 2.1$\times {10}^{3}$ | |

${F}_{a}$ | 9.92 | 10 | 10 | 10 | 9.60 | 10 | 9.79 | 10 | |

$S{D}_{f}$ | 0.27 | 0 | 0 | 0 | 0.24 | 0 | 0.24 | 0 | |

${T}_{c}$=$\lceil \ell /4\rceil $ | ${N}_{r}$ | 30 | 30 | 30 | 30 | ||||

${N}_{e}$ | 4.7$\times {10}^{4}$ | 4.0$\times {10}^{4}$ | 4.3$\times {10}^{4}$ | 6.2$\times {10}^{4}$ | |||||

$S{D}_{e}$ | 7.1$\times {10}^{4}$ | 3.5$\times {10}^{4}$ | 1.4$\times {10}^{4}$ | 6.8$\times {10}^{4}$ | |||||

${F}_{a}$ | 10 | 10 | 10 | 10 | |||||

$S{D}_{f}$ | 0 | 0 | 0 | 0 | |||||

${T}_{c}$=$\lceil \ell /8\rceil $ | ${N}_{r}$ | 30 | 30 | 30 | 30 | ||||

${N}_{e}$ | 1.2$\times {10}^{5}$ | 1.2$\times {10}^{5}$ | 1.5$\times {10}^{5}$ | 1.6$\times {10}^{5}$ | |||||

$S{D}_{e}$ | 1.0$\times {10}^{5}$ | 0.8$\times {10}^{5}$ | 0.8$\times {10}^{5}$ | 0.8$\times {10}^{5}$ | |||||

${F}_{a}$ | 10 | 10 | 10 | 10 | |||||

$S{D}_{f}$ | 0 | 0 | 0 | 0 | |||||

HDEP (ℓ = 30) | |||||||||

P = 4 | P = 60 | P = 80 | P = 100 | P = 2000 | P = 2000 | P = 5000 | P = 200 | ||

${T}_{c}$=$\lceil \ell /2\rceil $ | ${N}_{r}$ | 6 | 27 | 28 | 30 | 0 | 30 | 0 | 30 |

${N}_{e}$ | 4.7$\times {10}^{5}$ | 3.0$\times {10}^{5}$ | 1.1$\times {10}^{5}$ | 6.1$\times {10}^{4}$ | - | 1.4$\times {10}^{5}$ | - | 7.4$\times {10}^{3}$ | |

$S{D}_{e}$ | 5.1$\times {10}^{5}$ | 9.1$\times {10}^{5}$ | 2.6$\times {10}^{5}$ | 1.4$\times {10}^{4}$ | - | 6.5$\times {10}^{4}$ | - | 2.0$\times {10}^{3}$ | |

${F}_{a}$ | 9.74 | 9.97 | 9.97 | 10 | 9.46 | 10 | 9.69 | 10 | |

$S{D}_{f}$ | 0.13 | 0.08 | 0.07 | 0 | 0.20 | 0 | 0.01 | 0 | |

${T}_{c}$=$\lceil \ell /4\rceil $ | ${N}_{r}$ | 8 | 22 | 25 | 26 | ||||

${N}_{e}$ | 9.3$\times {10}^{5}$ | 1.5$\times {10}^{5}$ | 2.1$\times {10}^{5}$ | 1.1$\times {10}^{5}$ | |||||

$S{D}_{e}$ | 5.1$\times {10}^{5}$ | 3.0$\times {10}^{5}$ | 5.2$\times {10}^{5}$ | 2.2$\times {10}^{5}$ | |||||

${F}_{a}$ | 9.77 | 9.91 | 9.94 | 9.95 | |||||

$S{D}_{f}$ | 0.13 | 0.13 | 0.11 | 0.10 | |||||

${T}_{c}$=$\lceil \ell /8\rceil $ | ${N}_{r}$ | 21 | 28 | 26 | 30 | ||||

${N}_{e}$ | 9.1$\times {10}^{5}$ | 3.0$\times {10}^{5}$ | 4.3$\times {10}^{5}$ | 3.3$\times {10}^{5}$ | |||||

$S{D}_{e}$ | 11.0$\times {10}^{5}$ | 2.7$\times {10}^{5}$ | 5.4$\times {10}^{5}$ | 1.6$\times {10}^{5}$ | |||||

${F}_{a}$ | 9.91 | 9.97 | 9.95 | 10 | |||||

$S{D}_{f}$ | 0.13 | 0.07 | 0.10 | 0 | |||||

HDEP (ℓ = 40) | |||||||||

P = 4 | P = 60 | P = 80 | P = 100 | P = 2000 | P = 2000 | P = 5000 | P = 200 | ||

${T}_{c}$=$\lceil \ell /2\rceil $ | ${N}_{r}$ | 4 | 17 | 19 | 16 | 0 | 19 | 0 | 30 |

${N}_{e}$ | 9.4$\times {10}^{5}$ | 7.4$\times {10}^{4}$ | 2.1$\times {10}^{5}$ | 1.5$\times {10}^{5}$ | - | 2.5$\times {10}^{6}$ | - | 1.1$\times {10}^{4}$ | |

$S{D}_{e}$ | 7.1$\times {10}^{5}$ | 2.1$\times {10}^{4}$ | 4.7$\times {10}^{5}$ | 1.0$\times {10}^{5}$ | - | 4.5$\times {10}^{5}$ | - | 0.03$\times {10}^{4}$ | |

${F}_{a}$ | 9.26 | 9.75 | 9.82 | 9.81 | 9.02 | 9.72 | 9.15 | 10 | |

$S{D}_{f}$ | 0.30 | 0.29 | 0.24 | 0.21 | 0.04 | 0.37 | 0.04 | 0 | |

${T}_{c}$=$\lceil \ell /4\rceil $ | ${N}_{r}$ | 5 | 11 | 10 | 11 | ||||

${N}_{e}$ | 2.3$\times {10}^{6}$ | 1.4$\times {10}^{6}$ | 3.9$\times {10}^{5}$ | 4.1$\times {10}^{5}$ | |||||

$S{D}_{e}$ | 0.7$\times {10}^{6}$ | 1.4$\times {10}^{6}$ | 9.0$\times {10}^{5}$ | 5.9$\times {10}^{5}$ | |||||

${F}_{a}$ | 9.32 | 9.55 | 9.61 | 9.65 | |||||

$S{D}_{f}$ | 0.34 | 0.37 | 0.31 | 0.31 | |||||

${T}_{c}$=$\lceil \ell /8\rceil $ | ${N}_{r}$ | 4 | 3 | 6 | 7 | ||||

${N}_{e}$ | 2.7$\times {10}^{6}$ | 4.5$\times {10}^{5}$ | 1.2$\times {10}^{6}$ | 1.8$\times {10}^{6}$ | |||||

$S{D}_{e}$ | 1.3$\times {10}^{6}$ | 3.9$\times {10}^{5}$ | 1.3$\times {10}^{6}$ | 1.0$\times {10}^{6}$ | |||||

${F}_{a}$ | 9.37 | 9.42 | 9.43 | 9.43 | |||||

$S{D}_{f}$ | 0.30 | 0.26 | 0.33 | 0.35 |

EDT | SGA | CBGA | ecGA | LTGA | |||||
---|---|---|---|---|---|---|---|---|---|

NKL-K4 (ℓ = 20) | |||||||||

P = 4 | P = 60 | P = 80 | P = 100 | P = 2000 | P = 2000 | P = 5000 | P = 300 | ||

${T}_{c}$=$\lceil \ell /2\rceil $ | ${N}_{r}$ | 22 | 28 | 30 | 30 | 28 | 30 | 30 | 30 |

${N}_{e}$ | 1.4$\times {10}^{6}$ | 1.2$\times {10}^{4}$ | 1.5$\times {10}^{4}$ | 2.4$\times {10}^{4}$ | 1.2$\times {10}^{5}$ | 2.7$\times {10}^{4}$ | 5.7$\times {10}^{4}$ | 2.2$\times {10}^{3}$ | |

$S{D}_{e}$ | 1.4$\times {10}^{6}$ | 0.5$\times {10}^{4}$ | 0.7$\times {10}^{4}$ | 0.8$\times {10}^{4}$ | 1.1$\times {10}^{5}$ | 1.8$\times {10}^{4}$ | 3.4$\times {10}^{4}$ | 1.7$\times {10}^{3}$ | |

${F}_{a}$ | 0.6552 | 0.6585 | 0.6591 | 0.6591 | 0.6585 | 0.6591 | 0.6591 | 0.6591 | |

$S{D}_{f}$ | 0.0079 | 0.0035 | 0 | 0 | 0.0035 | 0 | 0 | 0 | |

${T}_{c}$=$\lceil \ell /4\rceil $ | ${N}_{r}$ | 30 | 30 | 30 | 30 | ||||

${N}_{e}$ | 2.5$\times {10}^{5}$ | 1.1$\times {10}^{5}$ | 8.9$\times {10}^{4}$ | 8.2$\times {10}^{4}$ | |||||

$S{D}_{e}$ | 6.0$\times {10}^{5}$ | 3.9$\times {10}^{5}$ | 18.5$\times {10}^{4}$ | 23.5$\times {10}^{4}$ | |||||

${F}_{a}$ | 0.6591 | 0.6591 | 0.6591 | 0.6591 | |||||

$S{D}_{f}$ | 0 | 0 | 0 | 0 | |||||

${T}_{c}$=$\lceil \ell /8\rceil $ | ${N}_{r}$ | 30 | 30 | 30 | 30 | ||||

${N}_{e}$ | 1.1$\times {10}^{5}$ | 8.6$\times {10}^{4}$ | 9.8$\times {10}^{4}$ | 7.0$\times {10}^{4}$ | |||||

$S{D}_{e}$ | 0.9$\times {10}^{5}$ | 8.2$\times {10}^{4}$ | 9.3$\times {10}^{4}$ | 6.8$\times {10}^{4}$ | |||||

${F}_{a}$ | 0.6591 | 0.6591 | 0.6591 | 0.6591 | |||||

$S{D}_{f}$ | 0 | 0 | 0 | 0 | |||||

NKL-K4 (ℓ = 30) | |||||||||

P = 4 | P = 60 | P = 80 | P = 100 | P = 2000 | P = 2000 | P = 5000 | P = 300 | ||

${T}_{c}$=$\lceil \ell /2\rceil $ | ${N}_{r}$ | 9 | 16 | 15 | 19 | 1 | 26 | 4 | 19 |

${N}_{e}$ | 1.0$\times {10}^{6}$ | 4.1$\times {10}^{4}$ | 5.3$\times {10}^{4}$ | 6.2$\times {10}^{4}$ | 1.5$\times {10}^{6}$ | 1.7$\times {10}^{6}$ | 1.6$\times {10}^{5}$ | 1.6$\times {10}^{4}$ | |

$S{D}_{e}$ | 1.4$\times {10}^{6}$ | 1.1$\times {10}^{4}$ | 0.8$\times {10}^{4}$ | 1.1$\times {10}^{4}$ | - | 1.2$\times {10}^{6}$ | 0.4$\times {10}^{5}$ | 0.4$\times {10}^{4}$ | |

${F}_{a}$ | 0.6498 | 0.6530 | 0.6526 | 0.6543 | 0.6456 | 0.6574 | 0.6422 | 0.6543 | |

$S{D}_{f}$ | 0.0061 | 0.0065 | 0.0065 | 0.0063 | 0.0032 | 0.0044 | 0.0088 | 0.0063 | |

${T}_{c}$=$\lceil \ell /4\rceil $ | ${N}_{r}$ | 21 | 17 | 20 | 17 | ||||

${N}_{e}$ | 1.2$\times {10}^{6}$ | 5.6$\times {10}^{5}$ | 1.2$\times {10}^{6}$ | 6.8$\times {10}^{5}$ | |||||

$S{D}_{e}$ | 1.4$\times {10}^{6}$ | 9.5$\times {10}^{5}$ | 1.2$\times {10}^{6}$ | 8.9$\times {10}^{5}$ | |||||

${F}_{a}$ | 0.6552 | 0.6585 | 0.6591 | 0.6591 | |||||

$S{D}_{f}$ | 0.0059 | 0.0064 | 0.0061 | 0.0064 | |||||

${T}_{c}$=$\lceil \ell /8\rceil $ | ${N}_{r}$ | 12 | 24 | 24 | 23 | ||||

${N}_{e}$ | 2.0$\times {10}^{6}$ | 1.3$\times {10}^{6}$ | 1.6$\times {10}^{6}$ | 1.3$\times {10}^{6}$ | |||||

$S{D}_{e}$ | 1.0$\times {10}^{6}$ | 1.1$\times {10}^{6}$ | 1.3$\times {10}^{6}$ | 1.0$\times {10}^{6}$ | |||||

${F}_{a}$ | 0.6505 | 0.6565 | 0.6565 | 0.6561 | |||||

$S{D}_{f}$ | 0.0076 | 0.0052 | 0.0052 | 0.0055 | |||||

NKL-K4 (ℓ = 40) | |||||||||

P = 4 | P = 60 | P = 80 | P = 100 | P = 2000 | P = 2000 | P = 5000 | P = 200 | ||

${T}_{c}$=$\lceil \ell /2\rceil $ | ${N}_{r}$ | 2 | 4 | 6 | 6 | 0 | 11 | 0 | 1 |

${N}_{e}$ | 5.0$\times {10}^{5}$ | 9.4$\times {10}^{4}$ | 1.1$\times {10}^{5}$ | 1.1$\times {10}^{5}$ | - | 1.9$\times {10}^{6}$ | - | 1.6$\times {10}^{4}$ | |

$S{D}_{e}$ | 2.2$\times {10}^{5}$ | 2.5$\times {10}^{4}$ | 0.3$\times {10}^{5}$ | 0.3$\times {10}^{5}$ | - | 1.3$\times {10}^{3}$ | - | - | |

${F}_{a}$ | 0.6495 | 0.6503 | 0.6507 | 0.6513 | 0.6366 | 0.6529 | 0.6354 | 0.6447 | |

$S{D}_{f}$ | 0.0031 | 0.0035 | 0.0044 | 0.0039 | 0.0084 | 0.0047 | 0.0080 | 0.0049 | |

${T}_{c}$=$\lceil \ell /4\rceil $ | ${N}_{r}$ | 5 | 2 | 5 | 4 | ||||

${N}_{e}$ | 1.3$\times {10}^{6}$ | 6.8$\times {10}^{4}$ | 6.3$\times {10}^{5}$ | 1.1$\times {10}^{6}$ | |||||

$S{D}_{e}$ | 1.2$\times {10}^{6}$ | 0.1$\times {10}^{4}$ | 11.3$\times {10}^{5}$ | 1.4$\times {10}^{6}$ | |||||

${F}_{a}$ | 0.6503 | 0.6489 | 0.6505 | 0.6501 | |||||

$S{D}_{f}$ | 0.0046 | 0.0032 | 0.0040 | 0.0038 | |||||

${T}_{c}$=$\lceil \ell /8\rceil $ | ${N}_{r}$ | 2 | 5 | 4 | 5 | ||||

${N}_{e}$ | 2.8$\times {10}^{6}$ | 1.9$\times {10}^{6}$ | 1.1$\times {10}^{6}$ | 7.9$\times {10}^{5}$ | |||||

$S{D}_{e}$ | 0.05$\times {10}^{6}$ | 1.4$\times {10}^{6}$ | 1.4$\times {10}^{6}$ | 8.8$\times {10}^{5}$ | |||||

${F}_{a}$ | 0.6456 | 0.6499 | 0.6493 | 0.6495 | |||||

$S{D}_{f}$ | 0.0071 | 0.0044 | 0.0044 | 0.0048 |

EDT | SGA | CBGA | LTGA | |||||
---|---|---|---|---|---|---|---|---|

MKP ($\alpha $ = 0.75) (ℓ = 100) | ||||||||

P = 4 | P = 60 | P = 80 | P = 100 | P = 2000 | P = 2000 | P = 400 | ||

${T}_{c}$=$\lceil \ell /2\rceil $ | ${N}_{r}$ | 1 | 0 | 0 | 0 | 0 | 1 | 0 |

${N}_{e}$ | 3.2$\times {10}^{6}$ | - | - | - | - | 9.6$\times {10}^{4}$ | - | |

$S{D}_{e}$ | - | - | - | - | - | - | - | |

${F}_{a}$ | 59,878 | 59,895 | 59,912 | 59,916 | 58,888 | 59,960 | 58,709 | |

$S{D}_{f}$ | 70 | 40 | 37 | 44 | 170 | 0.89 | 198 | |

${T}_{c}$=$\lceil \ell /4\rceil $ | ${N}_{r}$ | 2 | 0 | 1 | 1 | |||

${N}_{e}$ | 2.0$\times {10}^{6}$ | - | 1.3$\times {10}^{6}$ | 3.7$\times {10}^{5}$ | ||||

$S{D}_{e}$ | 0.8$\times {10}^{6}$ | - | - | - | ||||

${F}_{a}$ | 59,665 | 59,901 | 59,911 | 59,912 | ||||

$S{D}_{f}$ | 78 | 62 | 52 | 55 | ||||

${T}_{c}$=$\lceil \ell /8\rceil $ | ${N}_{r}$ | 0 | 0 | 1 | 0 | |||

${N}_{e}$ | - | - | 8.9$\times {10}^{5}$ | - | ||||

$S{D}_{e}$ | - | - | - | - | ||||

${F}_{a}$ | 59,665 | 59,901 | 59,911 | 59,912 | ||||

$S{D}_{f}$ | 138 | 51 | 55 | 41 | ||||

MKP ($\alpha $ = 0.50) (ℓ = 100) | ||||||||

P = 4 | P = 60 | P = 80 | P = 100 | P = 2000 | P = 2000 | P = 400 | ||

${T}_{c}$=$\lceil \ell /2\rceil $ | ${N}_{r}$ | 0 | 1 | 1 | 0 | 0 | 0 | 0 |

${N}_{e}$ | - | 6.6$\times {10}^{5}$ | 2.7$\times {10}^{6}$ | - | - | - | - | |

$S{D}_{e}$ | - | - | - | - | - | - | - | |

${F}_{a}$ | 44,454 | 44,474 | 44,466 | 44,467 | 43,717 | 44,509 | 43,385 | |

$S{D}_{f}$ | 51 | 54 | 46 | 46 | 150 | 18 | 205 | |

${T}_{c}$=$\lceil \ell /4\rceil $ | ${N}_{r}$ | 1 | 0 | 0 | 1 | |||

${N}_{e}$ | 2.1$\times {10}^{6}$ | - | - | 3.0$\times {10}^{6}$ | ||||

$S{D}_{e}$ | - | - | - | - | ||||

${F}_{a}$ | 44,464 | 44,468 | 44,472 | 44,466 | ||||

$S{D}_{f}$ | 54 | 54 | 47 | 53 | ||||

${T}_{c}$=$\lceil \ell /8\rceil $ | ${N}_{r}$ | 0 | 1 | 0 | 0 | |||

${N}_{e}$ | - | 2.8$\times {10}^{6}$ | - | - | ||||

$S{D}_{e}$ | - | - | - | - | ||||

${F}_{a}$ | 44,420 | 44,477 | 44,463 | 44,484 | ||||

$S{D}_{f}$ | 77 | 54 | 49 | 43 | ||||

MKP ($\alpha $ = 0.25) (ℓ = 100) | ||||||||

P = 4 | P = 60 | P = 80 | P = 100 | P = 2000 | P = 2000 | P = 300 | ||

${T}_{c}$=$\lceil \ell /2\rceil $ | ${N}_{r}$ | 2 | 1 | 2 | 4 | 0 | 0 | 0 |

${N}_{e}$ | 1.6$\times {10}^{6}$ | 3.7$\times {10}^{6}$ | 2.3$\times {10}^{5}$ | 2.6$\times {10}^{5}$ | - | - | - | |

$S{D}_{e}$ | 1.3$\times {10}^{6}$ | - | 0.1$\times {10}^{5}$ | 0.5$\times {10}^{5}$ | - | - | - | |

${F}_{a}$ | 24,309 | 24,323 | 24,331 | 24,340 | 24,316 | 24,343 | 23,984 | |

$S{D}_{f}$ | 52 | 44 | 47 | 46 | 33 | 4 | 123 | |

${T}_{c}$=$\lceil \ell /4\rceil $ | ${N}_{r}$ | 1 | 1 | 1 | 1 | |||

${N}_{e}$ | 3.3$\times {10}^{5}$ | 8.2$\times {10}^{4}$ | 1.6$\times {10}^{5}$ | 3.1$\times {10}^{5}$ | ||||

$S{D}_{e}$ | - | - | - | - | ||||

${F}_{a}$ | 24,315 | 24,323 | 24,317 | 24,334 | ||||

$S{D}_{f}$ | 44 | 49 | 43 | 41 | ||||

${T}_{c}$=$\lceil \ell /8\rceil $ | ${N}_{r}$ | 1 | 0 | 1 | 0 | |||

${N}_{e}$ | 3.2$\times {10}^{6}$ | - | 2.1$\times {10}^{6}$ | - | ||||

$S{D}_{e}$ | - | - | - | - | ||||

${F}_{a}$ | 24,314 | 24,326 | 24,324 | 24,310 | ||||

$S{D}_{f}$ | 48 | 46 | 43 | 42 |

EDT | SGA | CBGA | LTGA | |
---|---|---|---|---|

HIFF (ℓ = 32) | (3,3,2) | (4,4,3) | (1,1,4) | (1,1,1) |

HIFF (ℓ = 64) | (3,3,2) | (4,4,3) | (2,2,3) | (1,1,1) |

HIFF (ℓ = 128) | (2,2,2) | (2,4,2) | (2,3,2) | (1,1,1) |

Average | 2.444 | 3.333 | 2.222 | 1 |

HTRAP (ℓ = 27) | (2,3,2) | (3,2,4) | (1,1,3) | (1,1,1) |

HTRAP (ℓ = 81) | (2,4,2) | (3,2,3) | (3,2,3) | (1,1,1) |

Average | 2.5 | 2.833 | 2.166 | 1 |

HDEP (ℓ = 20) | (1,1,2) | (4,4,3) | (1,1,4) | (1,1,1) |

HDEP (ℓ = 30) | (1,1,2) | (4,4,4) | (1,1,3) | (1,1,1) |

HDEP (ℓ = 40) | (2,2,2) | (4,4,4) | (2,3,3) | (1,1,1) |

Average | 1.555 | 3.888 | 2.111 | 1 |

NKL-K4 (ℓ = 20) | (1,1,2) | (4,4,4) | (1,1,3) | (1,1,1) |

NKL-K4 (ℓ = 30) | (2,2,2) | (4,4,3) | (1,1,4) | (3,3,1) |

NKL-K4 (ℓ = 40) | (2,2,2) | (4,4,4) | (1,1,3) | (3,3,1) |

Average | 1.777 | 3.888 | 1.777 | 1.888 |

MKP (ℓ = 100, $\alpha $ = 0.75) | (1,2,2) | (3,3,3) | (2,1,1) | (3,4,3) |

MKP (ℓ = 100, $\alpha $ = 0.50) | (1,2,1) | (2,3,2) | (2,1,2) | (2,4,2) |

MKP (ℓ = 100, $\alpha $ = 0.25) | (1,2,1) | (2,3,2) | (2,1,2) | (2,4,2) |

Average | 1.444 | 2.555 | 1.555 | 2.888 |

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## Share and Cite

**MDPI and ACS Style**

Ohnishi, K.; Hamano, K.; Koeppen, M.
An Empirical Investigation on Evolutionary Algorithm Evolving Developmental Timings. *Electronics* **2020**, *9*, 1866.
https://doi.org/10.3390/electronics9111866

**AMA Style**

Ohnishi K, Hamano K, Koeppen M.
An Empirical Investigation on Evolutionary Algorithm Evolving Developmental Timings. *Electronics*. 2020; 9(11):1866.
https://doi.org/10.3390/electronics9111866

**Chicago/Turabian Style**

Ohnishi, Kei, Kouta Hamano, and Mario Koeppen.
2020. "An Empirical Investigation on Evolutionary Algorithm Evolving Developmental Timings" *Electronics* 9, no. 11: 1866.
https://doi.org/10.3390/electronics9111866