# Approaching the Pareto Front in a Biobjective Bus Route Design Problem Dealing with Routing Cost and Individuals’ Walking Distance by Using a Novel Evolutionary Algorithm

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Problem Description

## 4. HEABBR: A Hybrid Evolutionary Algorithm for the BBRP

#### 4.1. Chromosome Encoding

#### 4.2. Construction of a Feasible Solution

#### 4.2.1. Tentative Solution

- if $i\in W$ and ${b}_{i}=1$, the pickup point i is inserted in the route which provides the minimum insertion cost. At this time, i will be a visited pickup point.
- If $i\in W$ and ${b}_{i}=0$, the pickup point is discarded.
- If $i\in U$, the individual located at node i is allocated to the pickup point visited by a route that provides the minimum walking distance, bearing in mind the bus capacity constraint.

#### 4.2.2. Common Local Search Procedure: Reducing the Walking Distance

#### 4.2.3. Common Local Search Procedure: 2-opt

#### 4.2.4. Specific Local Search Procedure: Changing the Role of Nodes

#### Cycle Reduction

#### Cycle Augmentation

#### Termination Condition for the Specific Local Search Procedure

#### 4.2.5. Common Local Search Procedure: Removing Unused Pickup Point Locations and Combining Routes

#### 4.3. Initial Population

#### 4.4. Crossover and Mutation Operators

#### 4.5. Fitness Evaluation and Population Handling

## 5. Computational Experiment

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

BBRP | Biobjective Bus Routing Problem |

BRP | Bus Routing Problem |

CVRP | Capacitated Vehicle Routing Problem |

GRASP + VND | Greedy Randomized Adaptive Search + Variable Neighborhood Descent |

HEABBR | Hybrid Evolutionary Algorithm for the BBR problem |

NSGA-II | Nondominated Sorting Genetic Algorithm-II |

PISA | A Platform and Programming Language Independent Interface for |

Search Algorithms | |

SBRP | School Bus Routing Problem |

## References

- Ehrgott, M. Multicriteria Optimization, 2nd ed.; Springer: Berlin/Heildeberg, Germany, 2005. [Google Scholar]
- Deb, K. Multi-Objective Optimization Using Evolutionary Algorithms; Wiley: Chichester, UK, 2001. [Google Scholar]
- Coello Coello, C.A.; González Brambila, S.; Figueroa Gamboa, J.; Castillo Tapia, M.G.; Hernández Gómez, R. Evolutionary multiobjective optimization: Open research areas and some challenges lying ahead. Complex Intell. Syst.
**2020**, 6, 221–236. [Google Scholar] [CrossRef] [Green Version] - Ehrgott, M.; Gandibleux, X. Approximative solution methods for multiobjective combinatorial optimization. Top
**2004**, 12, 1–89. [Google Scholar] [CrossRef] - Jozefowiez, N.; Semet, F.; Talbi, E.G. Multi-objective vehicle routing problems. Eur. J. Oper. Res.
**2008**, 189, 293–309. [Google Scholar] [CrossRef] - Potvin, J.Y. State-of-the-art review—Evolutionary algorithms for vehicle routing. INFORMS J. Comput.
**2009**, 21, 517–656. [Google Scholar] [CrossRef] - Newton, R.M.; Thomas, W. Designing of school bus routes by computer. Socio-Econ. Plan. Sci.
**1969**, 3, 75–85. [Google Scholar] [CrossRef] - Braekers, K.; Ramaekers, K.; Van Nieuwenhuyse, I. The vehicle routing problem: State of the art classification and review. Comput. Ind. Eng.
**2016**, 99, 300–313. [Google Scholar] [CrossRef] - Eksioglu, B.; Volkan Vural, A.; Reisman, A. The vehicle routing problem: A taxonomic review. Comput. Ind. Eng.
**2009**, 57, 1472–1483. [Google Scholar] [CrossRef] - Tan, S.Y.; Yeh, W.C. Vehicle Routing Problem: State-of-the-Art Classification and Review. Appl. Sci.
**2021**, 11, 295. [Google Scholar] [CrossRef] - Park, J.; Kim, B.I. The school bus routing problem: A review. Eur. J. Oper. Res.
**2010**, 202, 311–319. [Google Scholar] [CrossRef] - Ellegood, W.A.; Solomon, S.; North, J.; Campbell, J.F. School bus routing problem: Contemporary trends and research directions. Omega
**2020**, 95, 102056. [Google Scholar] [CrossRef] - Peker, G.; Eliiyi, D.T. Shuttle bus service routing: A systematic literature review. Pamukkale Univ. J. Eng. Sci.
**2022**, 28, 160–172. [Google Scholar] - Corberán, A.; Fernández, E.; Laguna, M.; Martí, R. Heuristic solutions to the problem of routing school buses with multiple objectives. J. Oper. Res. Soc.
**2002**, 53, 427–435. [Google Scholar] [CrossRef] - Pacheco, J.; Caballero, R.; Laguna, M.; Molina, J. Bi-objective bus routing: An application to school buses in rural areas. Transp. Sci.
**2013**, 47, 397–411. [Google Scholar] [CrossRef] - Dasdemir, E.; Testik, M.C.; Öztürk, D.T.; Şakar, C.T.; Güleryüz, G.; Testik, O.M. A multi-objective open vehicle routing problem with overbooking: Exact and heuristic solution approaches for an employee transportation problem. Omega
**2022**, 108, 102587. [Google Scholar] [CrossRef] - Dulac, G.; Ferland, J.A.; Forgues, P.A. School bus routes generator in urban surroundings. Comput. Oper. Res.
**1980**, 7, 199–213. [Google Scholar] [CrossRef] - Chapleau, L.; Ferland, J.A.; Rousseau, J.M. Clustering for routing in densely populated areas. Eur. J. Oper. Res.
**1985**, 20, 48–57. [Google Scholar] [CrossRef] - Bowerman, R.; Hall, B.; Calamai, P. A multi-objective optimization approach to urban school bus routing: Formulation and solution method. Transp. Res. Part A Policy Pract.
**1995**, 29, 107–123. [Google Scholar] [CrossRef] - Riera-Ledesma, J.; Salazar-González, J.J. Solving school bus routing using the multiple vehicle traveling purchaser problem: A branch-and-cut approach. Comput. Oper. Res.
**2012**, 39, 391–404. [Google Scholar] [CrossRef] - Riera-Ledesma, J.; Salazar-González, J.J. A column generation approach for a school bus routing problem with resource constraints. Comput. Oper. Res.
**2013**, 40, 566–583. [Google Scholar] [CrossRef] - Schittekat, P.; Kinable, J.; Sörensen, K.; Sevaux, M.; Spieksma, F.; Springael, J. A metaheuristic for the school bus routing problem with bus stop selection. Eur. J. Oper. Res.
**2013**, 229, 518–528. [Google Scholar] [CrossRef] - Kinable, J.; Spieksma, F.C.R.; Berghe, G.V. School bus routing-a column generation approach. Int. Trans. Oper. Res.
**2014**, 21, 453–478. [Google Scholar] [CrossRef] [Green Version] - Calvete, H.I.; Galé, C.; Iranzo, J.A.; Toth, P. A partial allocation local search matheuristic for solving the school bus routing problem with bus stop selection. Mathematics
**2020**, 8, 1214. [Google Scholar] [CrossRef] - Bögl, M.; Doerner, K.F.; Parragh, S.N. The school bus routing and scheduling problem with transfers. Networks
**2015**, 65, 180–203. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Calvete, H.I.; Galé, C.; Iranzo, J.A.; Toth, P. The bilevel school bus routing problem with student choice: A bilevel approach and a simple and effective metaheuristic. Int. Trans. Oper. Res.
**2021**. [Google Scholar] [CrossRef] - Affenzeller, M.; Wagner, S.; Winkler, S.; Beham, A. Genetic Algorithms and Genetic Programming: Modern Concepts and Practical Applications; Chapman & Hall/CRC: London, UK, 2009. [Google Scholar]
- Chion, R.; Weise, T.; Michalewicz, Z. (Eds.) Variants of Evolutionary Algorithms for Real-World Applications; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Michalewick, Z. Genetic Algorithms + Data Structures = Evolution Programs, 3rd ed.; Springer: Berlin/Heidelberg, Germany, 1996. [Google Scholar]
- Calvete, H.I.; Galé, C.; Iranzo, J.A. An evolutionary algorithm for the biobjective capacitated m-ring star problem. In Algorithmic Decision Theory AFT 2013; Lecture Notes in Artificial Intelligence; Perny, P., Pirlot, M., Tsoukiàs, A., Eds.; Springer: Berlin/Heidelberg, Germany, 2013; Volume 8176, pp. 116–129. [Google Scholar]
- Calvete, H.I.; Galé, C.; Iranzo, J.A. MEALS: A multiobjective evolutionary algorithm with local search for solving the bi-objective ring star problem. Eur. J. Oper. Res.
**2016**, 250, 377–388. [Google Scholar] [CrossRef] - Fischetti, M.; Salazar-González, J.J.; Toth, P. A branch-and-cut algorithm for the symmetric generalized traveling salesman problem. Oper. Res.
**1997**, 45, 378–394. [Google Scholar] [CrossRef] - Croes, G.A. A method for solving traveling-salesman problems. Oper. Res.
**1958**, 6, 791–812. [Google Scholar] [CrossRef] - Deb, K.; Pratap, A.; Agrawal, S.; Meyarivan, T. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput.
**2002**, 6, 182–197. [Google Scholar] [CrossRef] [Green Version] - MacDonald, D.T. C++ Implementation of the Transportation Simplex Algorithm. Available online: https://github.com/engine99/transport-simplex (accessed on 29 March 2022).
- Zitzler, E.; Thiele, L. Multiobjective evolutionary algorithms: A comparative case study and the strength pareto approach. IEEE Trans. Evol. Comput.
**1999**, 3, 257–271. [Google Scholar] [CrossRef] [Green Version] - Zitzler, E.; Thiele, L.; Laumanns, M.; Fonseca, C.M.; Grunert da Fonseca, V. Performance assessment of multiobjective optimizers: An analysis and review. IEEE Trans. Evol. Comput.
**2003**, 7, 117–132. [Google Scholar] [CrossRef] [Green Version] - Bleuler, S.; Laumanns, M.; Thiele, L.; Zitzler, E. PISA—A Platform and Programming Language Independent Interface for Search Algorithms. In Evolutionary Multi-Criterion Optimization (EMO 2003); Lecture Notes in Computer Science; Fonseca, C.M., Fleming, P.J., Zitzler, E., Deb, K., Thiele, L., Eds.; Springer: Berlin/Heidelberg, Germany, 2003; pp. 494–508. [Google Scholar]

**Figure 1.**Two feasible solutions with the same potential pickup points, the same number of routes and different costs. The red circle represents the depot. The red square represents the destination. Blue squares are the potential pickup points and black circles are the individuals. The bus capacity is equal to eight.

**Figure 2.**Image in the objective space of a set of feasible solutions. The Pareto front is shown in red.

**Figure 5.**Value of the corresponding indicator for the instances in each set and configuration in blue. The red ball shows the mean value of the corresponding set instances.

**Figure 8.**Network and Pareto front of instances 35 and 78. The green square represents the depot. The red squares represent the pickup points. The black circles are the individuals.

Set | ID | Number of Pickup Points | Computing Time (min) |
---|---|---|---|

${S}_{1}$ | 1 to 24 | 5 | 2 |

${S}_{2}$ | 25 to 48 | 10 | 4 |

${S}_{3}$ | 49 to 72 | 20 | 8 |

${S}_{4}$ | 73 to 96 | 40 | 16 |

${S}_{5}$ | 97 to 112 | 80 | 32 |

Conf. | Crossover | Chromosome Update |
---|---|---|

1 | Single point | No update |

2 | Single point | Update |

3 | Uniform | No update |

4 | Uniform | Update |

**Table 3.**Aggregated results by a set of instances (indicators ${I}_{H}^{-}$ and ${I}_{{\u03f5}^{+}}^{1}$ have been multiplied by 100).

Conf. | Set | Size of the Pareto Front | ${\mathit{I}}_{\mathit{H}}^{-}$ | ${\mathit{I}}_{{\mathit{\u03f5}}^{+}}^{1}$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Mean | Stdev | Max | Mean | Stdev | Max | Mean | Stdev | Max | ||

1 | ${S}_{1}$ | 10.83 | 3.92 | 20.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

${S}_{2}$ | 41.88 | 14.69 | 71.00 | 0.03 | 0.08 | 0.38 | 0.49 | 0.83 | 2.88 | |

${S}_{3}$ | 109.00 | 41.30 | 182.00 | 0.20 | 0.24 | 0.81 | 1.23 | 0.95 | 3.48 | |

${S}_{4}$ | 142.33 | 67.49 | 272.00 | 1.13 | 1.03 | 3.60 | 2.66 | 2.01 | 8.98 | |

${S}_{5}$ | 120.44 | 66.77 | 269.00 | 2.48 | 2.30 | 8.84 | 5.59 | 4.56 | 14.23 | |

2 | ${S}_{1}$ | 10.83 | 3.92 | 20.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

${S}_{2}$ | 40.46 | 13.49 | 69.00 | 0.32 | 0.57 | 2.53 | 1.68 | 2.28 | 9.11 | |

${S}_{3}$ | 108.54 | 41.55 | 185.00 | 0.82 | 1.39 | 6.54 | 2.54 | 2.81 | 11.78 | |

${S}_{4}$ | 149.13 | 70.18 | 274.00 | 1.80 | 1.26 | 4.31 | 4.00 | 2.36 | 8.32 | |

${S}_{5}$ | 139.56 | 73.05 | 268.00 | 2.18 | 1.38 | 5.55 | 4.16 | 2.04 | 8.09 | |

3 | ${S}_{1}$ | 10.83 | 3.92 | 20.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

${S}_{2}$ | 41.67 | 14.73 | 72.00 | 0.03 | 0.06 | 0.22 | 0.41 | 0.64 | 2.40 | |

${S}_{3}$ | 112.71 | 42.59 | 194.00 | 0.11 | 0.14 | 0.57 | 0.94 | 0.68 | 2.51 | |

${S}_{4}$ | 141.04 | 75.35 | 286.00 | 0.92 | 0.75 | 2.46 | 2.29 | 1.51 | 4.95 | |

${S}_{5}$ | 106.44 | 66.40 | 273.00 | 2.67 | 2.32 | 9.95 | 5.42 | 3.83 | 14.93 | |

4 | ${S}_{1}$ | 10.83 | 3.92 | 20.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

${S}_{2}$ | 40.71 | 13.73 | 70.00 | 0.34 | 0.61 | 2.66 | 1.73 | 2.27 | 9.11 | |

${S}_{3}$ | 109.50 | 43.18 | 190.00 | 0.78 | 1.45 | 6.85 | 2.27 | 2.73 | 11.78 | |

${S}_{4}$ | 139.42 | 73.79 | 265.00 | 2.03 | 1.53 | 6.13 | 4.32 | 2.72 | 11.33 | |

${S}_{5}$ | 113.38 | 68.08 | 244.00 | 3.06 | 1.32 | 5.51 | 5.73 | 2.04 | 8.53 |

**Table 4.**Configuration 3: Number of points in the Pareto front (Pf) and indicator ${I}_{H}^{-}$ and ${I}_{{\u03f5}^{+}}^{1}$ values (multiplied by 100).

Ins. | Pf | ${\mathit{I}}_{\mathit{H}}^{-}$ | ${\mathit{I}}_{{\mathit{\u03f5}}^{+}}^{1}$ | Ins. | Pf | ${\mathit{I}}_{\mathit{H}}^{-}$ | ${\mathit{I}}_{{\mathit{\u03f5}}^{+}}^{1}$ | Ins. | Pf | ${\mathit{I}}_{\mathit{H}}^{-}$ | ${\mathit{I}}_{{\mathit{\u03f5}}^{+}}^{1}$ | Ins. | Pf | ${\mathit{I}}_{\mathit{H}}^{-}$ | ${\mathit{I}}_{{\mathit{\u03f5}}^{+}}^{1}$ | Ins. | Pf | ${\mathit{I}}_{\mathit{H}}^{-}$ | ${\mathit{I}}_{{\mathit{\u03f5}}^{+}}^{1}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 17 | 0.00 | 0.00 | 25 | 47 | 0.00 | 0.00 | 49 | 134 | 0.07 | 1.67 | 73 | 202 | 0.17 | 1.08 | 97 | 97 | 4.78 | 12.08 |

2 | 11 | 0.00 | 0.00 | 26 | 55 | 0.00 | 0.00 | 50 | 178 | 0.07 | 0.87 | 74 | 274 | 0.14 | 0.72 | 98 | 273 | 0.43 | 1.67 |

3 | 16 | 0.00 | 0.00 | 27 | 59 | 0.01 | 0.45 | 51 | 167 | 0.02 | 0.84 | 75 | 268 | 0.42 | 1.02 | 99 | 117 | 2.05 | 5.36 |

4 | 10 | 0.00 | 0.00 | 28 | 36 | 0.00 | 0.00 | 52 | 169 | 0.00 | 0.00 | 76 | 286 | 0.12 | 0.70 | 100 | 234 | 0.43 | 1.22 |

5 | 9 | 0.00 | 0.00 | 29 | 38 | 0.00 | 0.00 | 53 | 109 | 0.11 | 1.04 | 77 | 197 | 0.43 | 1.49 | 101 | 125 | 1.12 | 1.82 |

6 | 11 | 0.00 | 0.00 | 30 | 32 | 0.00 | 0.00 | 54 | 143 | 0.00 | 0.39 | 78 | 243 | 0.11 | 0.67 | 102 | 161 | 2.17 | 5.72 |

7 | 9 | 0.00 | 0.00 | 31 | 35 | 0.02 | 1.10 | 55 | 88 | 0.03 | 0.66 | 79 | 113 | 0.27 | 0.83 | 103 | 100 | 2.70 | 8.31 |

8 | 16 | 0.00 | 0.00 | 32 | 37 | 0.00 | 0.00 | 56 | 73 | 0.00 | 0.14 | 80 | 113 | 0.07 | 0.60 | 104 | 121 | 0.26 | 0.99 |

9 | 20 | 0.00 | 0.00 | 33 | 72 | 0.01 | 0.83 | 57 | 133 | 0.57 | 2.51 | 81 | 130 | 0.77 | 1.87 | 105 | 63 | 4.28 | 7.24 |

10 | 14 | 0.00 | 0.00 | 34 | 56 | 0.14 | 1.17 | 58 | 159 | 0.17 | 1.86 | 82 | 233 | 0.69 | 1.54 | 106 | 66 | 1.60 | 3.40 |

11 | 11 | 0.00 | 0.00 | 35 | 64 | 0.01 | 0.66 | 59 | 126 | 0.29 | 2.22 | 83 | 129 | 0.38 | 1.20 | 107 | 46 | 2.80 | 4.26 |

12 | 10 | 0.00 | 0.00 | 36 | 43 | 0.00 | 0.00 | 60 | 194 | 0.02 | 0.69 | 84 | 175 | 1.22 | 2.01 | 108 | 70 | 2.96 | 4.77 |

13 | 11 | 0.00 | 0.00 | 37 | 50 | 0.00 | 0.00 | 61 | 122 | 0.09 | 1.10 | 85 | 102 | 0.73 | 2.16 | 109 | 57 | 2.86 | 5.64 |

14 | 10 | 0.00 | 0.00 | 38 | 48 | 0.08 | 1.01 | 62 | 132 | 0.01 | 0.28 | 86 | 159 | 1.78 | 4.95 | 110 | 48 | 9.95 | 14.93 |

15 | 11 | 0.00 | 0.00 | 39 | 35 | 0.00 | 0.00 | 63 | 76 | 0.02 | 0.38 | 87 | 75 | 2.02 | 4.26 | 111 | 55 | 1.87 | 5.74 |

16 | 9 | 0.00 | 0.00 | 40 | 55 | 0.00 | 0.00 | 64 | 78 | 0.04 | 0.31 | 88 | 104 | 0.21 | 0.98 | 112 | 70 | 2.50 | 3.64 |

17 | 6 | 0.00 | 0.00 | 41 | 19 | 0.00 | 0.00 | 65 | 86 | 0.07 | 1.30 | 89 | 76 | 2.00 | 3.69 | ||||

18 | 10 | 0.00 | 0.00 | 42 | 47 | 0.12 | 1.55 | 66 | 100 | 0.36 | 1.67 | 90 | 62 | 1.90 | 3.30 | ||||

19 | 7 | 0.00 | 0.00 | 43 | 19 | 0.00 | 0.00 | 67 | 57 | 0.01 | 0.24 | 91 | 104 | 0.81 | 3.74 | ||||

20 | 11 | 0.00 | 0.00 | 44 | 40 | 0.04 | 0.70 | 68 | 99 | 0.12 | 1.18 | 92 | 91 | 0.62 | 1.57 | ||||

21 | 5 | 0.00 | 0.00 | 45 | 21 | 0.00 | 0.00 | 69 | 51 | 0.09 | 1.09 | 93 | 76 | 1.57 | 2.91 | ||||

22 | 13 | 0.00 | 0.00 | 46 | 38 | 0.22 | 2.40 | 70 | 113 | 0.28 | 1.15 | 94 | 57 | 2.46 | 4.77 | ||||

23 | 2 | 0.00 | 0.00 | 47 | 13 | 0.00 | 0.00 | 71 | 30 | 0.00 | 0.15 | 95 | 53 | 1.40 | 4.39 | ||||

24 | 11 | 0.00 | 0.00 | 48 | 41 | 0.00 | 0.00 | 72 | 88 | 0.12 | 0.83 | 96 | 63 | 1.67 | 4.49 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 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**

Calvete, H.I.; Galé, C.; Iranzo, J.A.
Approaching the Pareto Front in a Biobjective Bus Route Design Problem Dealing with Routing Cost and Individuals’ Walking Distance by Using a Novel Evolutionary Algorithm. *Mathematics* **2022**, *10*, 1390.
https://doi.org/10.3390/math10091390

**AMA Style**

Calvete HI, Galé C, Iranzo JA.
Approaching the Pareto Front in a Biobjective Bus Route Design Problem Dealing with Routing Cost and Individuals’ Walking Distance by Using a Novel Evolutionary Algorithm. *Mathematics*. 2022; 10(9):1390.
https://doi.org/10.3390/math10091390

**Chicago/Turabian Style**

Calvete, Herminia I., Carmen Galé, and José A. Iranzo.
2022. "Approaching the Pareto Front in a Biobjective Bus Route Design Problem Dealing with Routing Cost and Individuals’ Walking Distance by Using a Novel Evolutionary Algorithm" *Mathematics* 10, no. 9: 1390.
https://doi.org/10.3390/math10091390