# Genetic Algorithm for Scheduling Optimization Considering Heterogeneous Containers: A Real-World Case Study

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## Abstract

**:**

## 1. Introduction

#### Problem Definition

- a set of n activities with their durations $D=\left({d}_{1},{d}_{2},{d}_{3},\dots ,{d}_{n}\right)$, and a set of infinite containers with their capacities $C=\left({c}_{1},{c}_{2},{c}_{3},\dots \right)$, we search for
- schedule with all n activities programmed to be performed in a container such that
- the capacity of the containers is not exceeded, and
- idle time is minimized.

- Inpatient: patients that have to stay overnight in the hospital [13].
- Outpatient: patients that do not stay overnight in the hospital [14].
- Emergency patients: patients that arrive due to emergencies [15].
- Online scheduling: patients without appointment are considered [16].
- Offline scheduling: only patients with appointment are considered [17].

## 2. A Brief Review of the Specialized Literature

## 3. Development

#### 3.1. Genetic Algorithm

#### 3.1.1. Genetic Encoding

#### 3.1.2. Fitness Function

#### 3.1.3. Initial Population

#### 3.1.4. Selection

#### 3.1.5. Crossover

#### First-Fit Decreasing (FFD)

#### Numerical Example

#### 3.1.6. Mutation

#### Numerical Example

#### 3.1.7. Stop Condition

- t is the summation of the duration of all activities to schedule.
- i is a control parameter, and
- ${c}_{i}$ is the capacity of the ith OR.

#### 3.2. Strategy for the Use of Heterogeneous ORs

#### Case Study

#### 3.3. Computational Complexity

^{2})), whilst merge sort was applied to large sets, i.e., all related sets to the input data, chiefly the number of surgeries (whose worst case is in $\mathrm{O}\left(n{\mathrm{log}}_{2}n\right)$).

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

## 4. Results

#### 4.1. Test Instances

#### 4.2. Experimental Conditions

#### 4.3. Results

#### 4.4. Results Comparison

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Goldberg, D.E. Genetic Algorithms in Search, Optimization, and Machine Learning; Addison-Wesley Professional: Boston, MA, USA, 1989. [Google Scholar]
- Agarwal, M.; Srivastava, G.M.S. A Genetic Algorithm Inspired Task Scheduling in Cloud Computing. In Proceedings of the IEEE International Conference on Computing, Communication and Automation(ICCCA), Greater Noida, India, 29–30 April 2016. [Google Scholar]
- Sheng, X.; Li, Q. Templeta-based Genetic Algorithm for QoS-aware Task Scheduling in Cloud Computing. In Proceedings of the 2016 International Conference on Advanced Cloud and BigData (CBD), Chengdu, China, 13–14 August 2016. [Google Scholar]
- Gao, X.M.; Yang, Y.; Wu, H.Z. Genetic algorithm for scheduling double different size crane system with different truck ready times. In Proceedings of the 2016 IEEE International Conference on Industrial Engineering and Engineering Management (IEEM), Bali, Indonesia, 4–7 December 2016. [Google Scholar]
- Marques, I.; Captivo, M.E. Bicriteria elective surgery scheduling using an evolutionary algorith. Oper. Res. Health Care
**2015**, 7, 14–26. [Google Scholar] [CrossRef] - Fügener, A.; Hans, E.W.; Kolisch, R.; Kortbeek, N.; Vanberkel, P. Master surgery scheduling with consideration of multiple downstream. Eur. J. Oper. Res.
**2014**, 239, 227–236. [Google Scholar] [CrossRef] - Ma, L.; Gong, M.; Lui, J.; Cai, Q.; Jiao, L. Multi-level learning based memetic algorithm for community detection. Appl. Soft Comput.
**2014**, 19, 121–133. [Google Scholar] [CrossRef] - Mia, L.; Li, J.; Lin, Q.; Gong, M.; Coello, C.A.C.; Zhong, M. Cost-Aware Robust Control of Signed Networks by Using a Memetic Algorithm. IEEE Trans. Cybern.
**2019**, 1–14. [Google Scholar] [CrossRef] [PubMed] - Quiroz-Castellanos, M.; Cruz-Reyes, L.; Torres-Jimenez, J.; Gómez, C.; Fraire-Huacuja, H.J.; Alvim, A.C. A grouping genetic algorithm with controlled gene transmission for the bin packing problem. Comput. Oper. Res.
**2015**, 55, 52–64. [Google Scholar] [CrossRef] - Rivera, G.; Rodas-Osollo, J.; Bañuelos, P.; Quiroz, M.; Lopez, M. A Genetic Algorithm for surgery Scheduling Optimization in a Mexican Public Hospital. In Recent Advances in Artificial Intelligence Research and Development; IOS Press: Amsterdam, The Netherlands, 2017; pp. 269–274. [Google Scholar]
- Conforti, D.; Guerriero, F.; Guido, R. A MultiObjetive Block Scheduling Model for the Managment of Surgical Operating Rooms: New Solution Approaches via Genetic Algorithms; Health Care Management (WHCM): Venice, Italy, 2010. [Google Scholar]
- Marques, I.; Captivo, M.E.; Pato, M.V. Scheduling elective surgeries in a Portuguese hospital using a genetic heuristic. Oper. Res. Health Care
**2014**, 3, 59–72. [Google Scholar] [CrossRef] - Zhao, L.; Chien, C.F.; Gren, M. A bi-objective genetic algorithm for intelligent rehabilitation scheduling considering therapy precedence constraints. J. Intell. Manuf.
**2018**, 29, 873–988. [Google Scholar] [CrossRef] - Azadeh, A.; Baghersad, M.; Farahani, M.H.; Zarrin, M. Semi-online patient scheduling in pathology laboratories. Artif. Intell. Med.
**2015**, 64, 217–226. [Google Scholar] [CrossRef] [PubMed] - Azadeh, A.; Farahani, M.H.; Torabzadeh, S.; Baghersad, M. Scheduling prioritized patients in emergency department laboratories. Comput. Methods Programs Biomed.
**2014**, 117, 61–70. [Google Scholar] [CrossRef] [PubMed] - Braaksma, A.; Kortbeek, N.; Post, G.F.; Nollet, F. Integral multidisciplinary rehabilitation treatment planning. Oper. Res. Health Care
**2014**, 3, 145–159. [Google Scholar] [CrossRef] [Green Version] - Bikker, I.A.; Kortbeek, N.; vanOs, R.M.; Boucherie, R.J. Reducing access times for radiation treatment by aligning the doctor’s schemes. Oper. Res. Health Care
**2015**, 7, 111–121. [Google Scholar] [CrossRef] - Marynissen, J.; Demeulemeester, E. Literature Review on Integrated Hospital Scheduling Problems; Technical Report; Faculty of Economics and Business, KU Leuven: Leuven, Belgium, 2016. [Google Scholar]
- Budylskiy, A.V.; Kvyatkovskaya, I.Y. Using coevolution genetic algorithm with Pareto principles to solve project scheduling problem under duration and cost constraints. J. Inf. Organ. Sci.
**2014**, 38, 1–9. [Google Scholar] - Ortiz-Aguilar, L.; Carpio-Valadez, J.M.; Puga-Soberanes, H.J.; Díaz-González, C.L.; Lino-Ramirez, C.; Soria-Alcaraz, J.A. Comparativa de algoritmos bioinspirados aplicados al problema de calendarización de horarios [Comparison of bioinspired algorithms applied to the schedule scheduling problem]. Res. Comput. Sci.
**2015**, 94, 33–43. [Google Scholar] - Xiang, W.; Yin, J.; Lim, G. An ant colony optimization approach for solving an operating room surgery scheduling problem. Comput. Ind. Eng.
**2015**, 85, 335–345. [Google Scholar] [CrossRef] - Marchesi, J.F.; Cavalcanti, M.A. A Genetic Algorithm Approach for the Master Surgical Schedule Problem; IEEE conference on Evolving and Adaptive Intelligent Systems (EAIS); IEEE: Natal, Brazil, 2016. [Google Scholar]
- Li, X.; Gao, L. An effective hybrid genetic algorithm and tabu search for flexible job shop scheduling problem. Int. J. Prod. Econ.
**2016**, 174, 93–110. [Google Scholar] [CrossRef] - Latorre-Nuñez, G.; Lüer-Villagra, A.; Marianov, V.; Obreque, C.; Ramis, F.; Neriz, L. Scheduling operating rooms with consideration of all resources, postanesthesia beds and emergency surgeries. Comput. Ind. Eng.
**2016**, 97, 248–257. [Google Scholar] - Chen-Yang, C.; Kuo-Ching, Y.; Hsia-Hsiang, C.; Jia-Xian, L. Optimization algorithms for proportionate flow shop scheduling problems with variable maintenance activities. Comput. Ind. Eng.
**2018**, 117, 164–170. [Google Scholar] [CrossRef] - Hosseinabadi, A.A.R.; Vahidi, J.; Saemi, B.; Sangaiah, A.K.; Elhoseny, M. Extended Genetic Algorithm for solving open-shop scheduling problem. Soft Comput.
**2019**, 23, 5099–5116. [Google Scholar] [CrossRef] - Guo, C.; Wang, C.; Zuo, X. A Genetic Algorithm based Column Generation Method for Multi-Depot Electric Bus Vehicle Scheduling. In Proceedings of the Genetic and Evolutionary Computation Conference Companion (GECCO2019), Prague, Czech Republic, 13–17 July 2019. [Google Scholar]
- Mahdi, S.; Fontes, D.M.; Fontes, F.C. A BRK GA for the Integrated Scheduling Problem in FMSs. In Proceedings of the Genetic and Evolutionary Computation Conference Companion (GECCO19), Prague, Czech Republic, 13–17 July 2019. [Google Scholar]
- Falkenauer, E.; Delchambre, A. A genetic algorithm for bin packing and line balancing. In Proceedings of the IEEE International Conference on Robotics and Automation, Nice, France, 12–14 May 1992. [Google Scholar]

**Figure 2.**Construction of a solution, (

**a**) FF-ñ applied to a set of surgeries; (

**b**) FF applied to the rest of the surgeries.

Instance | Number of Surgeries | Average Length of Surgery | Longest Surgery | Shortest Surgery |
---|---|---|---|---|

1 | 238 | 103 min | 420 min | 45 min |

2 | 210 | 125 min | 395 min | 40 min |

3 | 192 | 122 min | 380 min | 40 min |

4 | 221 | 124 min | 510 min | 40 min |

Parameter | Value |
---|---|

Population size | 100 |

Elite population size | 10 |

Maximum number of generations | 500 |

Solution lifespan | 10 |

Individuals for selection | 80 |

Individuals for mutation | 17 |

Instance 1 | Instance 2 | Instance 3 | Instance 4 | Average | ||||||
---|---|---|---|---|---|---|---|---|---|---|

GA [1] | GAfSO | GA | GAfSO | GA | GAfSO | GA | GAfSO | GA | GAfSO | |

Fitness | 96.33 | 95.54 | 95.2 | 94.4 | 96.1 | 95.5 | 94.26 | 96.04 | 94.81 | 95.3 |

Number of ORs used in the solution | 50 | 50 | 54 | 54 | 48 | 48 | 56 | 56 | 52 | 52 |

Number of days used in the solution | 12.5 | 12.5 | 13.5 | 13.5 | 12 | 12 | 14 | 14 | 13 | 13 |

Number of executions | 5 | 1 | 5 | 1 | 4 | 1 | 5 | 1 | 4.75 | 1 |

Average computational time (s) | 0.03 | 0.1 | 0.012 | 0.08 | 0.035 | 0.107 | 0.043 | 0.169 | 0.02 | 0.11 |

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**MDPI and ACS Style**

Rivera, G.; Cisneros, L.; Sánchez-Solís, P.; Rangel-Valdez, N.; Rodas-Osollo, J.
Genetic Algorithm for Scheduling Optimization Considering Heterogeneous Containers: A Real-World Case Study. *Axioms* **2020**, *9*, 27.
https://doi.org/10.3390/axioms9010027

**AMA Style**

Rivera G, Cisneros L, Sánchez-Solís P, Rangel-Valdez N, Rodas-Osollo J.
Genetic Algorithm for Scheduling Optimization Considering Heterogeneous Containers: A Real-World Case Study. *Axioms*. 2020; 9(1):27.
https://doi.org/10.3390/axioms9010027

**Chicago/Turabian Style**

Rivera, Gilberto, Luis Cisneros, Patricia Sánchez-Solís, Nelson Rangel-Valdez, and Jorge Rodas-Osollo.
2020. "Genetic Algorithm for Scheduling Optimization Considering Heterogeneous Containers: A Real-World Case Study" *Axioms* 9, no. 1: 27.
https://doi.org/10.3390/axioms9010027