# Multi-Objective Optimization of Experiments Using Curvature and Fisher Information Matrix

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## Abstract

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## 1. Introduction

## 2. Model-Based Optimal Design of Experiments

#### 2.1. Multi-Objective Design of Experiments Based on Curvatures

#### 2.2. Numerical Implementation of the Curvature-Based MOO Design

## 3. Results

#### 3.1. MBDOEs of Baker Yeast Fermentation Model

#### 3.2. Performance Evaluation

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

MBDOE | Model-based design of experiments |

FIM | Fisher information matrix |

MOO | Multi-objective optimization |

RMS | Root mean square |

ODE | Ordinary differential equation |

MLE | Maximum likelihood estimator |

CVP | Control vector parametrization |

nMSE | Normalized mean-square error |

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**Figure 1.**Iterative model identification cycle. The model building process involves the following key steps: experimental design, model structure formulation, parameter estimation, and model validation.

**Figure 2.**Control vector parametrization of input profiles. In the baker yeast case study, we implemented piecewise constant input profiles with ${u}_{i}$ = [${u}_{i,1}$, ${u}_{i,2}$, ${u}_{i,3}$, ${u}_{i,4}$, ${u}_{i,5}$] and four switching times: ${t}_{sw1}$, ${t}_{sw2}$, ${t}_{sw3}$, and ${t}_{sw4}$.

**Figure 3.**Optimal dilution factor and feed substrate concentration. Optimal dilution factor (${u}_{1}$ in h${}^{-1}$, left panels) and feed substrate concentration (${u}_{2}$ in g/L, right panels). (

**A**,

**B**) D-optimal (blue). (

**C**,

**D**) A-optimal (red). (

**E**,

**F**) E-optimal (green). (

**G**,

**H**) modified E-optimal (black). In panels (

**A**–

**H**), the optimal ${u}_{1}$ and ${u}_{2}$ using Fisher information matrix (FIM)-based criteria are shown by solid line. Those using FIM-based criteria combined with curvatures are shown by dashed line, while those using FIM-based criteria combined with parameter correlation are drawn with dashed-dot line. (

**I**–

**J**) Threshold curvature (magenta, solid line), and Q-optimal design (magenta, dashed line).

**Figure 4.**Optimal sampling grid from model-based design of experiments (MBDOEs). Simple Fisher information matrix (FIM)-based criteria shown by continuous line, FIM-based criteria combined with curvatures by dashed line, and FIM-based criteria combined with parameter correlation by dashed-pointed line. Dots indicate the sampling times.

FIM-Based MBDOE | Criterion |
---|---|

D-optimal | max ${\prod}_{i}{\lambda}_{i}$ |

A-optimal | max ${\sum}_{i}{\lambda}_{i}$ |

E-optimal | max $min\left({\lambda}_{i}\right)$ |

Modified E-optimal | max $\frac{min\left({\lambda}_{i}\right)}{max\left({\lambda}_{i}\right)}$ |

**Table 2.**Optimal initial condition of biomass ${x}_{1}\left(0\right)$ (g/L) from model-based design of experiments (MOO: multi-objective optimization).

Design Criterion | ${\mathit{x}}_{1}\left(0\right)$ |
---|---|

D-optimal | 10.0 |

MOO D-optimal and curvatures | 10.0 |

MOO D-optimal and correlation | 10.0 |

A-optimal | 10.0 |

MOO A-optimal and curvatures | 9.9 |

MOO A-optimal and correlation | 10.0 |

E-optimal | 10.0 |

MOO E-optimal and curvatures | 10.0 |

MOO E-optimal and correlation | 10.0 |

Modified E-optimal | 10.0 |

MOO modified E-optimal and curvatures | 10.0 |

MOO modified E-optimal and correlation | 10.0 |

Threshold curvature | 8.2 |

Q-optimal | 5.5 |

**Table 3.**Model-based design of experiment (MBDOE) performance on the fed-batch fermentation of baker’s yeast model. The overall parameter accuracy is represented by the average of the normalized mean-square error (nMSE). The reported parameter values and errors are the averages and standard deviations from 100 repeated runs of parameter estimation.

Design Criterion | $\overline{\mathrm{nMSE}}$ | ${\mathit{\theta}}_{1}$ ± ${\mathit{SD}}_{{\mathit{\theta}}_{1}}$ | ${\mathit{\theta}}_{2}$ ± ${\mathit{SD}}_{{\mathit{\theta}}_{2}}$ | ${\mathit{\theta}}_{3}$ ± ${\mathit{SD}}_{{\mathit{\theta}}_{3}}$ | ${\mathit{\theta}}_{4}$ ± ${\mathit{SD}}_{{\mathit{\theta}}_{4}}$ |
---|---|---|---|---|---|

D-optimal | 7.06 × 10${}^{-3}$ | 0.3107 ± 0.0102 | 0.1831 ± 0.0276 | 0.5505 ± 0.0125 | 0.0502 ± 0.0026 |

MOO D-optimal and curvatures | 4.71 × 10${}^{-3}$ | 0.3099 ± 0.0056 | 0.1825 ± 0.0233 | 0.5496 ± 0.0099 | 0.0499 ± 0.0018 |

MOO D-optimal and correlation | 5.36 × 10${}^{-3}$ | 0.3117 ± 0.0134 | 0.1781 ± 0.0151 | 0.5543 ± 0.0270 | 0.0508 ± 0.0049 |

A-optimal | 2.35 × 10${}^{-1}$ | 0.3294 ± 0.0659 | 0.2399 ± 0.1387 | 0.5841 ± 0.1083 | 0.0558 ± 0.0181 |

MOO A-optimal and curvatures | 1.42 | 0.3669 ± 0.0947 | 0.5267 ± 0.2230 | 0.5548 ± 0.1333 | 0.0510 ± 0.0244 |

MOO A-optimal and correlation | 4.82 | 0.0863 ± 0.0499 | 0.8927 ± 0.2555 | 0.2879 ± 0.1928 | 0.0177 ± 0.0263 |

E-optimal | 8.01 × 10${}^{-2}$ | 0.3180 ± 0.0420 | 0.2026 ± 0.0956 | 0.5473 ± 0.0159 | 0.0496 ± 0.0026 |

MOO E-optimal and curvatures | 3.33 × 10${}^{-3}$ | 0.3083 ± 0.0095 | 0.1829 ± 0.0164 | 0.5502 ± 0.0183 | 0.0500 ± 0.0026 |

MOO E-optimal and correlation | 8.19 × 10${}^{-3}$ | 0.3108 ± 0.0164 | 0.1824 ± 0.0213 | 0.5552 ± 0.0304 | 0.0509 ± 0.0055 |

Modified E-optimal | 6.99 × 10${}^{-2}$ | 0.3137 ± 0.0165 | 0.1986 ± 0.0920 | 0.5498 ± 0.0144 | 0.0502 ± 0.0033 |

MOO modified E-optimal and curvatures | 3.44 × 10${}^{-4}$ | 0.3095 ± 0.0036 | 0.1789 ± 0.0034 | 0.5491 ± 0.0073 | 0.0500 ± 0.0013 |

MOO modified E-optimal and correlation | 2.27 × 10${}^{-3}$ | 0.3088 ± 0.0048 | 0.1820 ± 0.0160 | 0.5486 ± 0.0047 | 0.0496 ± 0.0013 |

Threshold curvature | 1.29 × 10${}^{-2}$ | 0.3144 ± 0.0307 | 0.1857 ± 0.0339 | 0.5500 ± 0.0155 | 0.0502 ± 0.0032 |

Q-optimal | 1.91 × 10${}^{-2}$ | 0.3085 ± 0.0178 | 0.1757 ± 0.0216 | 0.5514 ± 0.0236 | 0.0504 ± 0.0119 |

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**MDPI and ACS Style**

Manesso, E.; Sridharan, S.; Gunawan, R.
Multi-Objective Optimization of Experiments Using Curvature and Fisher Information Matrix. *Processes* **2017**, *5*, 63.
https://doi.org/10.3390/pr5040063

**AMA Style**

Manesso E, Sridharan S, Gunawan R.
Multi-Objective Optimization of Experiments Using Curvature and Fisher Information Matrix. *Processes*. 2017; 5(4):63.
https://doi.org/10.3390/pr5040063

**Chicago/Turabian Style**

Manesso, Erica, Srinath Sridharan, and Rudiyanto Gunawan.
2017. "Multi-Objective Optimization of Experiments Using Curvature and Fisher Information Matrix" *Processes* 5, no. 4: 63.
https://doi.org/10.3390/pr5040063