# Applying a Genetic Algorithm to a m-TSP: Case Study of a Decision Support System for Optimizing a Beverage Logistics Vehicles Routing Problem

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

_{ij}defines the distance between cities i and j, and x

_{ijk}for the salesman k from cities i to j, considering that each city is visited only once [21].

- Generate XY, an N × 2 matrix, where N is the number of places, in which it was chosen to generate a “random” matrix of N × N. Thus, the results of the solution will be later portrayed on a map, instead of having to insert a matrix with the coordinates of each place.
- Insert DMAT, which is the N × N matrix of distances or costs. Distances between locations were looked up on Google Maps and imported in Microsoft Excel matrices.
- NSALESMEN represents the number of vendors visiting the locations, which differ in this problem between one salesman on the large experimental routes, two salesmen on the routes in the North and South zones and four in the Central area.
- MINTOUR is the minimum number of places to be visited by each salesman, in this case 10 locations on all routes except Central Route which was defined as 15 as there are more places. It was through this input that the number of places to visit per day, for each salesman, was kept balanced.
- POPSIZE was set to 80, which represents the number of solutions with which the algorithm starts iterations and mutations in search of a better solution. This parameter must be, by obligation of the algorithm, divisible by 8.
- NUMITER is the maximum number of iterations that the algorithm will do, in search of the best solution, in which case it was defined as 5000.
- The remaining inputs, were kept unchanged, remaining by default.

## 3. Case Study

## 4. Model Formulation

#### 4.1. North Zone

#### 4.2. South Zone

#### 4.3. Central Zone

## 5. Analysis of Results

#### 5.1. North Zone

#### 5.2. South Zone

#### 5.3. Central Zone

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Clusters ^{(1)} | Key Points |
---|---|

Inguias (1) | 217 |

Belmonte (12) | 202, 57, 175 |

Gonçalo (4) | 125 |

Cruz da Pedra (1) | 253 |

Vale Formoso (3) | 213 |

Valhelhas (3) | 15 |

Verdelhos (6) | 69 |

Vale da Amoreira (1) | 212 |

Manteigas (11) | 136, 139 |

Penhas da Saúde (4) | 230, 169, 180 |

Penedos Altos (16) | 121, 145, 215, 73 |

Parque Industrial Canhoso (1) | 60 |

Canhoso (7) | 162 |

Teixoso (10) | 148, 229, 261 |

Gibraltar (1) | 103 |

Vila do Carvalho (5) | 52 |

^{(1)}Number of establishments associated with the cluster.

Clusters ^{(1)} | Key Points |
---|---|

Covilhã Sul (4) | 260, 54 |

Ferro (2) | 218 |

Alcaria (1) | 259 |

Dominguizo (1) | 254 |

Vales do Rio (2) | 199 |

Peso (1) | 120 |

Barco (3) | 131 |

Paul (12) | 79, 220 |

Casegas (1) | 44 |

Erada (3) | 101 |

Unhais da Serra (15) | 105, 126 |

Cortes do Meio (5) | 133 |

Bairro do Cabeço (4) | 75 |

Tortosendo (9) | 86, 118, 208 |

Casal da Serra (3) | 18 |

Parque Industrial do Tortosendo (3) | 267 |

Ponte Pedrinha (1) | 59 |

Boidobra (11) | 214, 83 |

Refúgio (4) | 243, 270 |

^{(1)}Number of establishments associated with the cluster.

Route | Key Points | Distance (km) |
---|---|---|

Long route | 0→121→73→215→145→52→162→60→148→229→261→103→217→202→57→175→125→253→213→15→69→212→139→136→230→169→180→0 | 126.30 |

1st route | 0→217→202→57→175→125→253→213→15→69→212→139→136→230→169→180→0 | 118.60 |

2nd route | 0→121→60→162→148→229→261→103→52→145→215→73→0 | 21.10 |

Route | Key Points | Distance (km) |
---|---|---|

Long route | 0→260→54→218→259→254→199→120→131→79→44→220→101→105→126→133→75→118→18→208→86→267→59→214→83→243→270→0 | 119.65 |

1st route | 0→260→54→214→218→259→254→199→120→131→79→44→220→101→105→126→133→0 | 100.50 |

2nd route | 0→270→243→83→208→18→118→75→267→59→86→0 | 31.95 |

Route | Key Points | Distance (km) |
---|---|---|

1st route | 0→196→161→188→16→42→4→50→159→93→82→106→166→219→102→87→100→32→8→266→95→0 | 8.24 |

2nd route | 0→158→66→1→67→155→89→216→21→209→144→150→236→235→192→70→46→108→0 | 15.40 |

3rd route | 0→264→167→41→142→26→184→34→182→244→156→272→68→138→137→143→2→174→0 | 5.99 |

4th route | 0→53→71→128→186→258→233→122→99→152→96→22→153→47→37→276→247→0 | 9.03 |

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**MDPI and ACS Style**

Gomes, D.E.; Iglésias, M.I.D.; Proença, A.P.; Lima, T.M.; Gaspar, P.D.
Applying a Genetic Algorithm to a m-TSP: Case Study of a Decision Support System for Optimizing a Beverage Logistics Vehicles Routing Problem. *Electronics* **2021**, *10*, 2298.
https://doi.org/10.3390/electronics10182298

**AMA Style**

Gomes DE, Iglésias MID, Proença AP, Lima TM, Gaspar PD.
Applying a Genetic Algorithm to a m-TSP: Case Study of a Decision Support System for Optimizing a Beverage Logistics Vehicles Routing Problem. *Electronics*. 2021; 10(18):2298.
https://doi.org/10.3390/electronics10182298

**Chicago/Turabian Style**

Gomes, David E., Maria Inês D. Iglésias, Ana P. Proença, Tânia M. Lima, and Pedro D. Gaspar.
2021. "Applying a Genetic Algorithm to a m-TSP: Case Study of a Decision Support System for Optimizing a Beverage Logistics Vehicles Routing Problem" *Electronics* 10, no. 18: 2298.
https://doi.org/10.3390/electronics10182298