# Multi-Objective Optimization for Controlling the Dynamics of the Diabetic Population

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## Abstract

**:**

## 1. Introduction

- (1)
- A multi-objective mathematical model for controlling the dynamics of the diabetic population is introduced;
- (2)
- A discretization of the proposed model is realized based on the trapezoidal rule and the Euler–Cauchy method (we demonstrate that this error is bounded);
- (3)
- Two multi-objective optimizers are used to solve the proposed multi-objective model;
- (4)
- As a first postprocessing phase, FFT convolution is used to clean the noise from the control;
- (5)
- As a second post-processing phase, two soft clustering methods are used to structure the Pareto front.

## 2. Related Works

## 3. Methodology Overview

**Notations**

**Model**

**Discretization**

**Smart local search**

**Post-processing**

**Performance evaluation**

## 4. Multi-Objective Diabetic Control Model

#### 4.1. Single-Goal Control Model

#### 4.2. Multi-Objective Control Model

- (a)
- The primary methods used were gradient descent algorithms [29];
- (b)
- Dual methods, which exploit convexity to calculate the gradient of the dual max–min, were also used [30];
- (c)
- The substitution and the decomposition Lagrange methods that introduce a copy variable to decompose the initial problem to two sub-problems [31]: the first one does not have any constraints (for which we can use gradient descent, among others) and the second one does not have objective functions (for which we can use back tracking methods, among others).

**Lemma**

**1.**

**Proof of Lemma**

**1.**

**Proposition**

**1.**

**Proof of the Proposition**

**1.**

**Theorem**

**1.**

**Proof of the Theorem**

**1:**

#### 4.3. Pareto Controls Characterization

## 5. Smart Algorithms

#### 5.1. Swarm Intelligence Optimizers

#### 5.1.1. Non-Dominated Sorting Genetic Algorithm II

#### 5.1.2. Multi-Objective Firefly Algorithm

- (a)
- Fireflies have the ability to attract other fireflies, no matter which sex they are.
- (b)
- The attraction is positively proportional to the brightness. If all the fireflies have nearly the same degree of brightness, then one or more fireflies are moving.
- (c)
- The luminosity of a firefly is calculated from the cost function.

#### 5.2. Soft Clustering Algorithms

#### 5.2.1. Gaussian Mixture Model (GMM)

#### 5.2.2. Fuzzy C-Means (FCM)

## 6. Experimental Results and Discussion

#### 6.1. NSGA-II Combined with Soft Clustering Methods

#### 6.2. MOFA Combined with Soft Clustering Methods

**Notes**:

- (a)
- Considering the experimental results shown in the Figure A1, Figure A2, Figure A3 and Figure A4, we notice that the two soft clustering methods FCM and GMM give approximately the same groups (considering a simple permutation), so we extended the same remarks, conclusions, and recommendations to the other method.
- (b)
- (c)

#### 6.3. Single-Objective vs. Multi-Objective on the Control of the Dynamics of the Diabetic Population Problem (C2D2P)

#### 6.4. Sensitivity of the Proposed System

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A2.**Compartments obtained using different controls produced via GMM applied to the Pareto front produced via NSGA-II.

**Figure A4.**Compartments obtained when introducing different controls, produced via GMM applied to the Pareto front produced via MOFA applied to the model (4).

**Figure A5.**Selection of the optimal number of clusters based on the silhouette criteria of groups obtained via FCM.

## Appendix B

**Figure A7.**Compartments obtained when introducing different controls, produced via FCM, applied to the Pareto front produced via NSGA-II, for which we added Gaussian noise from [0, 0.05].

**Figure A8.**Compartments obtained when introducing different controls, produced via FCM, applied to the Pareto front produced via NSGA-II, for which we added Gaussian noise from [0, 0.1].

**Figure A9.**Compartments obtained when introducing different controls, produced via FCM, applied to Pareto front produced via NSGA-II, for which we added Gaussian noise from [0, 0.15].

**Figure A10.**Compartments obtained when introducing different controls, produced via FCM applied to the Pareto front produced via NSGA-II, for which we added Gaussian noise from [0, 0.2].

**Figure A11.**Compartments obtained when introducing different controls, produced via FCM applied to the Pareto front produced via NSGA-II, for which we added Gaussian noise from [0, 0.25].

**Figure A12.**Compartments obtained when introducing different controls, produced via FCM, applied to the Pareto front produced via NSGA-II, for which we added Gaussian noise from [0, 0.3].

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**Figure 5.**(

**a**) The set of controls ${u}_{1}$ extracted from the Pareto front produced via NSGA-II applied to model (4). (

**b**) The set of controls ${u}_{2}$ extracted from the Pareto front produced via NSGA-II applied to model (4).

**Figure 6.**Control pairs, over 10 years, obtained via FCM applied to the Pareto front produced via NSGA-II.

**Figure 7.**Compartments obtained when introducing different controls, produced via FCM applied to the Pareto front produced via NSGA-II, in model (4).

**Figure 9.**(

**a**) The set of controls ${u}_{1}$ extracted from the Pareto front produced via MOFA applied to the model (4). (

**b**) The set of controls ${u}_{2}$ extracted from the Pareto front produced via MOFA applied to the model (4).

**Figure 11.**Compartments obtained when introducing different controls, produced via FCM applied to the Pareto front produced via MOFA, in the model (4).

Option [12] | Configuration |
---|---|

Crossover operator | New_indiv = indiv1 + rand × atio × (indiv2 − indiv1) |

Crossover ratio | 0.8 |

Number of iterations | 60 |

Mutation ratio | adaptive feasible |

Option [13] | Configuration |
---|---|

Maximum number of iterations | 1000 |

Swarm size | 25 |

Light absorption coefficient | 1 |

Attraction coefficient base value | 2 |

Mutation coefficient | 0.2 |

Mutation coefficient damping ratio | 0.98 |

Cluster 1 | Cluster 2 | Cluster 3 | Cluster 4 | ||
---|---|---|---|---|---|

Multi-objective vs. single-objective | u_{1} | 3% | 4% | 4% | 4% |

u_{2} | 14% | 6% | 18% | 11% |

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## Share and Cite

**MDPI and ACS Style**

El Moutaouakil, K.; El Ouissari, A.; Palade, V.; Charroud, A.; Olaru, A.; Baïzri, H.; Chellak, S.; Cheggour, M.
Multi-Objective Optimization for Controlling the Dynamics of the Diabetic Population. *Mathematics* **2023**, *11*, 2957.
https://doi.org/10.3390/math11132957

**AMA Style**

El Moutaouakil K, El Ouissari A, Palade V, Charroud A, Olaru A, Baïzri H, Chellak S, Cheggour M.
Multi-Objective Optimization for Controlling the Dynamics of the Diabetic Population. *Mathematics*. 2023; 11(13):2957.
https://doi.org/10.3390/math11132957

**Chicago/Turabian Style**

El Moutaouakil, Karim, Abdellatif El Ouissari, Vasile Palade, Anas Charroud, Adrian Olaru, Hicham Baïzri, Saliha Chellak, and Mouna Cheggour.
2023. "Multi-Objective Optimization for Controlling the Dynamics of the Diabetic Population" *Mathematics* 11, no. 13: 2957.
https://doi.org/10.3390/math11132957