# Fractional Calculus to Analyze Efficiency Behavior in a Balancing Loop in a System Dynamics Environment

^{*}

## Abstract

**:**

## 1. Introduction

^{2}values.

## 2. Materials and Methods

#### 2.1. Gamma Function

#### 2.2. Mittag-Leffler Function

#### 2.3. Caputo Derivative

**Definition**

**1.**

## 3. Methodology

#### 3.1. System Dynamics Model

#### 3.2. System of Differential Equations

#### 3.3. Solution Ordinary Differential Equation

#### 3.4. OEE Methodology

#### 3.4.1. Availability Factor

#### 3.4.2. Quality Factor

#### 3.4.3. Performance Factor

## 4. Results

#### 4.1. Solution of The Fractional Order Model Per Caputo

#### 4.2. Mathematical Demonstration of Fractional Functions

#### 4.3. Simulating Diverse Scenarios

#### 4.3.1. Simulation of Weaving Department Efficiencies

#### 4.3.2. Simulation of Basting Department Efficiencies

#### 4.4. Field Validation

#### 4.4.1. Simulation of Weaving Department Efficiencies

#### 4.4.2. Simulation of Basting Department Efficiencies

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- García, J.M. Teoría y Ejercicios Prácticos de Dinámica de Sistemas: Dinámica de Sistemas con VENSIM PLE; Nova Science Publishers, Inc.: Hauppauge, NY, USA, 2023. [Google Scholar]
- Aracil, J.; Gordillo, F. Dinámica de Sistemas; Alianza Editorial: Madrid, Spain, 1997; p. 20. [Google Scholar]
- De Leo, E.; Aranda, D.; Addati, G. Introducción a la Dinámica de Sistemas; no. 739, Serie Documentos de Trabajo; University of CEMA: Buenos Aires, Argentina, 2020. [Google Scholar]
- Spinel, A.V. Análisis de Armónicos en Redes de Distribución con Recursos Renovables Conectados a Través de Inversores; Universidad De Los Andes: Bogota, Colombia, 2023. [Google Scholar]
- Abid, A.; Kallel, I.; Sanchez-Medina, J.; Ayed, M. Parameters Sensitivity Analysis of Ant Colony Based Clustering: Application for Student Grouping in Collaborative Learning Environment. IEEE Access
**2023**, 12, 24751–24761. [Google Scholar] [CrossRef] - Cazcarro, I.; García-Gusano, D.; Iribarren, D.; Linares, P.; Romero, J.; Arocena, P.; Arto, I.; Banacloche, S.; Lechón, Y.; Miguel, L.; et al. Energy-socio-economic-environmental modelling for the EU energy and post-COVID-19 transitions. Sci. Total Environ.
**2022**, 805, 150329. [Google Scholar] [CrossRef] [PubMed] - Angerhofer, B.; Angelides, M. System dynamics modelling in supply chain management: Research review. In Proceedings of the 2000 Winter Simulation Conference Proceedings (Cat. No.00CH37165), Orlando, FL, USA, 10–13 December 2000. [Google Scholar]
- Zimmermann, N.; Curran, K. Dynamics of interdisciplinarity: A microlevel analysis of communication and facilitation in a group model—Building workshop. Syst. Dyn. Rev.
**2023**, 39, 336–370. [Google Scholar] [CrossRef] - Ortega, J.J.C.; Serrato, R.B.; Morales, R.A.L. Development of a system dynamics model based on Six Sigma methodology. Ing. Investig.
**2017**, 37, 80. [Google Scholar] [CrossRef] - Wang, Z.; Li, X.; Mao, Y.; Li, L.; Wang, X.; Lin, Q. Dynamic simulation of land use change and assessment of carbon storage based on climate change scenarios at the city level: A case study of Bortala, China. Ecol. Indic.
**2022**, 134, 108499. [Google Scholar] [CrossRef] - Zoghi, M.; Kim, S. Dynamic Modeling for Life Cycle Cost Analysis of BIM-Based Construction Waste Management. Sustainability
**2020**, 12, 2483. [Google Scholar] [CrossRef] - Liu, J.; Liu, Y.; Wang, X. An environmental assessment model of construction and demolition waste based on system dynamics: A case study in Guangzhou. Environ. Sci. Pollut. Res.
**2020**, 27, 37237–37259. [Google Scholar] [CrossRef] - De-Blas, I.; Miguel, L.; De-Castro, C. Modelos de evaluación integrada (IAMs) aplicados al cambio climático y la transición energética. DYNA-Ing. Ind.
**2021**, 96, 316. [Google Scholar] - Sánchez, J.B.; Serrato, R.B.; Bianchetti, M. Design and Development of a Mathematical Model for an Industrial Process, in a System Dynamics Environment. Appl. Sci.
**2022**, 12, 9855. [Google Scholar] [CrossRef] - Mojtahedzadeh, M.; Qureshi, H.I. Understanding End of Life Practices: Perspectives on Communication, Religion and Culture; Springer International Publishing: Berlin/Heidelberg, Germany, 1997; pp. 261–274. [Google Scholar]
- Mojtahedzadeh, M. Do parallel lines meet? How can pathway participation metrics and eigenvalue analysis produce similar results? Syst. Dyn. Rev.
**2008**, 24, 451–478. [Google Scholar] [CrossRef] - Gonçalves, P. Behavior Modes, Pathways and Overall Trajectories: Eigenvector and Eigenvalue Analysis of Dynamic Systems. SSRN Electron. J.
**2009**, 25, 35–62. [Google Scholar] [CrossRef] - Oliva, R. Structural dominance analysis of large and stochastic models. Syst. Dyn. Rev.
**2016**, 32, 26–51. [Google Scholar] [CrossRef] - Schoenberg, W.; Hayward, J.; Eberlein, R. Improving Loops that Matter. Syst. Dyn. Rev.
**2023**, 39, 140–151. [Google Scholar] [CrossRef] - Sánchez, J.M.B.; Serrato, R.B. Design and Development of an Optimal Control Model in System Dynamics through State-Space Representation. Appl. Sci.
**2023**, 13, 7154. [Google Scholar] [CrossRef] - Sterman, J. Business Dynamics; Irwin/McGraw-Hill: New York, NY, USA, 2010; p. 982. [Google Scholar]
- Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications; Elsevier: Amsterdam, The Netherlands, 1998. [Google Scholar]
- Mainardi, F. Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics; Springer: Vienna, Austria, 1997; pp. 291–348. [Google Scholar]
- Magin, R. Fractional Calculus in Bioengineering, Part 1. Critical Reviews™. Biomed. Eng.
**2004**, 32, 104. [Google Scholar] - Ramadevi, B.; Kasi, V.R.; Bingi, K. Hybrid LSTM-Based Fractional-Order Neural Network for Jeju Island’s Wind Farm Power Forecasting. Fractal Fract.
**2024**, 8, 149. [Google Scholar] [CrossRef] - Moumen, A.; Mennouni, A.; Bouye, M. A Novel Vieta–Fibonacci Projection Method for Solving a System of Fractional In-tegrodifferential Equations. Mathematics
**2023**, 11, 3985. [Google Scholar] [CrossRef] - Schiessel, H.; Metzler, R.; Blumen, A.; Nonnenmacher, T.F. Generalized viscoelastic models: Their fractional equations with solutions. J. Phys. A Math. Gen.
**1995**, 28, 6567–6584. [Google Scholar] [CrossRef] - Meral, F.C.; Royston, T.J.; Magin, R. Fractional calculus in viscoelasticity: An experimental study. Commun. Nonlinear Sci. Numer. Simul.
**2010**, 15, 939–945. [Google Scholar] [CrossRef] - Chen, W.; Sun, H.; Zhang, X.; Korošak, D. Anomalous diffusion modeling by fractal and fractional derivatives. Comput. Math. Appl.
**2010**, 59, 1754–1758. [Google Scholar] [CrossRef] - Guia, M.; Gomez, F.; Rosales, J. Lomé and the North-South Relations (1975–1984): From the “New International Economic Order” to a New Conditionality. In Europe in a Globalising World; Nomos: Baden-Baden, Germany, 2013; Volume 11, pp. 123–146. [Google Scholar]
- Leonardo, M.-J.; Pedro, L.-L.; Adán, F.-B.; Manuel, L.-H.J. Automatic blood vessel detection using fractional Hessian matrices. ECORFAN J.
**2022**, 6, 12–19. [Google Scholar] - Sengupta, S.; Ghosh, U.; Sarkar, S.; Das, S. Prediction of Ventricular Hypertrophy of Heart Using Fractional Calculus. J. Appl. Nonlinear Dyn.
**2020**, 9, 287–305. [Google Scholar] [CrossRef] - Baba, I.A.; Humphries, U.W.; Rihan, F.A. Role of Vaccines in Controlling the Spread of COVID-19: A Fractional-Order Model. Vaccines
**2023**, 11, 145. [Google Scholar] [CrossRef] [PubMed] - Ruby; Mandal, M. The geometrical and physical interpretation of fractional order derivatives for a general class of functions. Math. Methods Appl. Sci.
**2024**, 2024, 1–21. [Google Scholar] [CrossRef] - Podlubny, I. Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation. arXiv
**2001**, arXiv:math/0110241. [Google Scholar] - Heymans, N.; Podlubny, I. Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol. Acta
**2006**, 45, 765–771. [Google Scholar] [CrossRef] - Tarasov, V.E. No violation of the Leibniz rule. No fractional derivative. Commun. Nonlinear Sci. Numer. Simul.
**2013**, 18, 2945–2948. [Google Scholar] [CrossRef] - Micula, S. An iterative numerical method for fractional integral equations of the second kind. J. Comput. Appl. Math.
**2018**, 339, 124–133. [Google Scholar] [CrossRef] - Jafari, H.; Tuan, N.; Ganji, R. A new numerical scheme for solving pantograph type nonlinear fractional integro-differential equations. J. King Saud Univ.—Sci.
**2021**, 33, 101185. [Google Scholar] [CrossRef] - Sánchez-Muñoz, J. Hamilton y el descubrimiento de los Cuaterniones. Pensam. Matemático
**2011**, 7. [Google Scholar] - De Oliveira, E.C.; Machado, T. A review of definitions for fractional derivatives and integral. Math. Probl. Eng.
**2014**, 2014, 1–6. [Google Scholar] [CrossRef] - Khalil, R.; Al Horani, M.; Yousef, A.; Sababheh, M. A new definition of fractional derivative. J. Comput. Appl. Math.
**2014**, 264, 65–70. [Google Scholar] [CrossRef] - Caputo, M.; Mainardi, F. A new dissipation model based on memory mechanism. Pure Appl. Geophys.
**1971**, 91, 134–147. [Google Scholar] [CrossRef] - Caputo, M.; Fabrizio, M. Theory and Applications of Fractional Order Systems. Progr. Fract. Differ. Appl.
**2015**, 1, 73–85. [Google Scholar] - Atangana, A.; Baleanu, D. New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model. Therm. Sci.
**2016**, 20, 763–769. [Google Scholar] [CrossRef] - Artin, E. The Gamma Function; Courier Dover Publications: Mineola, NY, USA, 2015. [Google Scholar]
- Mittag-Leffler, G. Sur la représentation analytique d’une branche uniforme d’une fonction monogène: Seconde note. Acta Math.
**1903**, 24, 183–204. [Google Scholar] [CrossRef] - Wiman, A. Über die Nullstellen der Funktionen E a (x). Acta Math.
**1905**, 29, 217–234. [Google Scholar] [CrossRef] - Wiman, A. Über den Fundamentalsatz in der Teorie der Funktionen E a (x). Acta Math.
**1905**, 29, 191–201. [Google Scholar] [CrossRef] - Zarslan, M.A.; Yılmaz, B. The extended Mittag-Leffler function and its properties. J. Inequalities Appl.
**2014**, 2014, 1–10. [Google Scholar] [CrossRef] - Khan, M.A.; Ahmed, S. On some properties of the generalized Mittag-Leffler function. SpringerPlus
**2013**, 2, 337. [Google Scholar] [CrossRef] - Makris, N. The Fractional Derivative of the Dirac Delta Function and Additional Results on the Inverse Laplace Transform of Irrational Functions. Fractal Fract.
**2021**, 5, 18. [Google Scholar] [CrossRef]

State Variable | Notation of the Variable | Xd (Goals) | Adder | Differential Equation |
---|---|---|---|---|

Weaving | $X$ | $Xd1=72$ | $Xd-x$ | $Ed1=(Xd1-x)k1$ |

Basting | $Y$ | $Xd2=72$ | $Xd2-y$ | Ed$2=(\mathrm{Xd}2-$ y) k2 − Ed1 |

Day | 25 | 26 | 27 | 28 | 29 | 30 |
---|---|---|---|---|---|---|

Total time (min) | 690 | 690 | 690 | 690 | 690 | 630 |

Breaks (min) | 120 | 120 | 120 | 120 | 120 | 120 |

Maintenance stoppage (min) | 10 | 10 | 10 | 10 | 10 | 45 |

Shutdowns of machine records (min) | 20.43 | 29.21 | 36.23 | 20.78 | 32.45 | 30.27 |

(A) Planned time available (min) | 560 | 560 | 560 | 560 | 560 | 465 |

(B) Productive time (min) | 539.5 | 530.7 | 523.7 | 539.2 | 527.5 | 434.73 |

(B/A) Availability (%) | 96.35 | 94.78 | 93.53 | 96.29 | 94.21 | 93.49 |

Day | 25 | 26 | 27 | 28 | 29 | 30 |
---|---|---|---|---|---|---|

Total Time (min) | 100 | 100 | 100 | 100 | 100 | 90 |

Breaks (min) | 20 | 20 | 20 | 20 | 20 | 20 |

(A) Planned time available (min) | 80 | 80 | 80 | 80 | 80 | 70 |

(B) Productive time (min) | 79.26 | 79.35 | 76.55 | 78.41 | 79.95 | 69.73 |

(B/A) Availability (%) | 99.10 | 99.19 | 95.70 | 98.03 | 99.95 | 99.63 |

Day | 25 | 26 | 27 | 28 | 29 | 30 |
---|---|---|---|---|---|---|

(A) Actual production- total parts | 80 | 79 | 72 | 72 | 69 | 55 |

Contaminated canvases | 0 | 1 | 0 | 2 | 0 | 0 |

Overlay | 5 | 6 | 5 | 10 | 7 | 10 |

Non-functional canvases | 1 | 3 | 3 | 5 | 3 | 7 |

(B) Good pieces | 74 | 69 | 64 | 55 | 59 | 38 |

(B/A) Quality (%) | 92.50 | 87.34 | 88.89 | 76.39 | 85.51 | 69.09 |

Day | 25 | 26 | 27 | 28 | 29 | 30 |
---|---|---|---|---|---|---|

(A) Actual production-total parts | 75 | 64 | 72 | 79 | 88 | 81 |

(B) Good pieces | 75 | 64 | 72 | 78 | 88 | 80 |

(B/A) Quality (%) | 100 | 100 | 100 | 98.73 | 100 | 98.77 |

Day | 25 | 26 | 27 | 28 | 29 | 30 |
---|---|---|---|---|---|---|

(A) Planned production (total pieces) | 82.7 | 79.3 | 72.98 | 73.44 | 69.24 | 55.21 |

(B) Real production | 80 | 79 | 72 | 72 | 69 | 55 |

(B/A) Performance (%) | 96.70 | 99.59 | 98.66 | 98.04 | 99.65 | 99.63 |

Day | 25 | 26 | 27 | 28 | 29 | 30 |
---|---|---|---|---|---|---|

(A) Planned production (total pieces) | 78.0 | 78.1 | 75.3 | 83.7 | 96.5 | 84.2 |

(B) Real production | 75 | 64 | 72 | 79 | 88 | 81 |

(B/A) Performance (%) | 96.10 | 81.93 | 95.54 | 94.38 | 91.16 | 96.20 |

Day | 25 | 26 | 27 | 28 | 29 | 30 | OEE Weekly |
---|---|---|---|---|---|---|---|

OEE (Weaving) (%) | 86.1 | 82.4 | 82.0 | 72.1 | 80.2 | 64.3 | 77.90 |

OEE (Basting) (%) | 95.2 | 81.2 | 91.4 | 91.3 | 91.1 | 94.6 | 90.84 |

Scenarios | Category | Target (Pieces) | Efficiency |
---|---|---|---|

Scenario 1 | Optimal | 100 | $\alpha =0.9$ |

Very good | $\alpha =0.8$ | ||

Good | $\alpha =0.7$ | ||

Fair | $\alpha =0.6$ | ||

Poor | $\alpha =0.5$ | ||

Very poor | $\alpha =0.4$ | ||

Scenario 2 | Optimal | 70 | $\alpha =0.9$ |

Very good | $\alpha =0.8$ | ||

Good | $\alpha =0.7$ | ||

Fair | $\alpha =0.6$ | ||

Poor | $\alpha =0.5$ | ||

Very poor | $\alpha =0.4$ | ||

Scenario 3 | Optimal | 50 | $\alpha =0.9$ |

Very good | $\alpha =0.8$ | ||

Good | $\alpha =0.7$ | ||

Fair | $\alpha =0.6$ | ||

Poor | $\alpha =0.5$ | ||

Very poor | $\alpha =0.4$ |

Scenarios | Category | Target (Pieces) | Efficiency |
---|---|---|---|

Scenario 1 | Optimal | 100 | $\alpha =0.9$ |

Very good | $\alpha =0.8$ | ||

Good | $\alpha =0.7$ | ||

Fair | $\alpha =0.6$ | ||

Poor | $\alpha =0.5$ | ||

Very poor | $\alpha =0.4$ | ||

Scenario 2 | Optimal | 70 | $\alpha =0.9$ |

Very good | $\alpha =0.8$ | ||

Good | $\alpha =0.7$ | ||

Fair | $\alpha =0.6$ | ||

Poor | $\alpha =0.5$ | ||

Very poor | $\alpha =0.4$ | ||

Scenario 3 | Optimal | 50 | $\alpha =0.9$ |

Very good | $\alpha =0.8$ | ||

Good | $\alpha =0.7$ | ||

Fair | $\alpha =0.6$ | ||

Poor | $\alpha =0.5$ | ||

Very poor | $\alpha =0.4$ |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Barrios-Sánchez, J.M.; Baeza-Serrato, R.; Martínez-Jiménez, L.
Fractional Calculus to Analyze Efficiency Behavior in a Balancing Loop in a System Dynamics Environment. *Fractal Fract.* **2024**, *8*, 212.
https://doi.org/10.3390/fractalfract8040212

**AMA Style**

Barrios-Sánchez JM, Baeza-Serrato R, Martínez-Jiménez L.
Fractional Calculus to Analyze Efficiency Behavior in a Balancing Loop in a System Dynamics Environment. *Fractal and Fractional*. 2024; 8(4):212.
https://doi.org/10.3390/fractalfract8040212

**Chicago/Turabian Style**

Barrios-Sánchez, Jorge Manuel, Roberto Baeza-Serrato, and Leonardo Martínez-Jiménez.
2024. "Fractional Calculus to Analyze Efficiency Behavior in a Balancing Loop in a System Dynamics Environment" *Fractal and Fractional* 8, no. 4: 212.
https://doi.org/10.3390/fractalfract8040212