# Comparison of Multi-Objective Evolutionary Algorithms to Solve the Modular Cell Design Problem for Novel Biocatalysis

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Multi-Objective Modular Cell Design

#### 2.2. Optimal Solutions for a Multi-Objective Optimization Problem

#### 2.3. MOEA Selection

#### 2.4. Performance Metrics

#### 2.5. Algorithm Parameters

#### 2.6. Metabolic Models

#### 2.7. Implementation

## 3. Results and Discussion

#### 3.1. Case 1: A 3-Objectives Design Problem

#### 3.2. Case 2: A 10-Objectives Design Problem

#### 3.3. Case 3: Use of Large Population Size Overcomes Poor MOEA Performance

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**–

**d**) Conceptual illustration of performance metrics of MOEAs for a two-objectives design problem. $PF$ and $P{F}^{\ast}$ correspond to the Pareto front approximation and the best Pareto front available, respectively. The reference point R must always dominate all solutions in $PF$. (

**e**–

**f**) An example of Pareto fronts with 2 dimensions and associated metrics. The 4th generation corresponds to ${\mathit{PF}}^{\ast}$ used as a reference for comparison.

**Figure 2.**Comparison of MOEAs for a 3-objectives design problem. (

**a**) The simplified metabolic pathways for conversion of glucose to the target products. Reducing equivalents are presented with ${e}^{-}$. (

**b**–

**l**) Generation-dependent performance metrics for various MOEAs. (

**m**) Performance metrics for various MOEAs at the last generation. (

**n**–

**x**) Pareto fronts of various MOEAs at the last generation. It should be noted that only the first replicate is plotted for clear illustration. (

**y**) Reference Pareto front (${\mathit{PF}}^{\ast}$). Each line represents a solution.

**Figure 3.**Comparison of MOEAs for a 10-objectives design problem. (

**a**–

**k**) Generation-dependent performance metrics for various MOEAs. (

**l**) Performance metrics for various MOEAs at the last generation.

**Figure 4.**Comparison of MOEAs for a 10-objectives design problem with larger population sizes (

**a**–

**k**) Generation-dependent performance metrics for various MOEAs. (

**l**) Performance metrics for various MOEAs at the last generation.

**Figure 5.**Wall-clock run times for the 10-objectives design problem with population sizes of 100 (Case 2) and 1000 (Case 3).

Abbreviation | Name | Notes | Reference |
---|---|---|---|

NSGAII | Non-dominated sorting genetic algorithm 2 | Highly applied MOEA | [23] |

gamultiobj | Matlab implementation of NSGAII | Used in the original ModCell2 study [8] | [21] |

MOEAIGDNS | Multi-objective evolutionary algorihtm based on an enhanced inverted generational distance metric | General MOEA with an implementation that works well with discrete variables | [29] |

ARMOEA | Adapation to reference points multi-objective evolutionary algorithm | Many-objective EA based on MOEAIGDNS | [30] |

EFRRR | Ensemble fitness ranking with ranking restriction | Many-objective EA | [31] |

MaOEADDFC | Many-objective evolutionary algorithm based on directional diversity and favorable convergence | Many-objective EA | [32] |

SPEAR | Strength Pareto evolutionary algorithm based on reference direction | Many-objective EA | [33] |

tDEA | $\theta $-dominance evolutionary algorithm | Many-objective EA | [34] |

BiGE | Bi-goal evolution | Many-objective EA | [35] |

NSGAIII | Non-dominated sorting genetic algorithm 3 | Many-objective EA | [36] |

SPEA2SDE | Strength Pareto evolutionary algorithm 2 with shift-based density estimation | Many-objective EA | [37] |

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**MDPI and ACS Style**

Garcia, S.; Trinh, C.T. Comparison of Multi-Objective Evolutionary Algorithms to Solve the Modular Cell Design Problem for Novel Biocatalysis. *Processes* **2019**, *7*, 361.
https://doi.org/10.3390/pr7060361

**AMA Style**

Garcia S, Trinh CT. Comparison of Multi-Objective Evolutionary Algorithms to Solve the Modular Cell Design Problem for Novel Biocatalysis. *Processes*. 2019; 7(6):361.
https://doi.org/10.3390/pr7060361

**Chicago/Turabian Style**

Garcia, Sergio, and Cong T. Trinh. 2019. "Comparison of Multi-Objective Evolutionary Algorithms to Solve the Modular Cell Design Problem for Novel Biocatalysis" *Processes* 7, no. 6: 361.
https://doi.org/10.3390/pr7060361