# Optimal Integrated Single-Framework Algorithm for the Multi-Level School Bus Network Problem

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Model

**Parameters:**

- ${\mathit{I}}_{\mathit{s}}^{\mathit{t}}$
**:**Number of students of school**s**of (t = 1: Elementary school, t = 2: Middle school, t = 3: High school) - ${\mathit{o}}_{\mathit{k}}:origin\text{}of\text{}bus\text{}\mathit{k}\text{}in\text{}morning$
**K**: Total number of buses- ${\mathit{n}}^{\mathit{t}}$: Number of schools (t = 1: Elementary school, t = 2: Middle school, t = 3: High school)
- ${\mathbf{d}}_{\mathbf{i}\mathbf{j}}^{\mathbf{s}}$ = direct distance between the home of student
**i**and the home of student**j**of school**s** - ${\mathbf{d}}_{{\mathbf{i}\mathbf{o}}_{\mathbf{k}}}^{\mathbf{s}}$ = direct distance between student
**i**of school**s**and start location of bus**k** - ${\mathit{d}\mathit{i}\mathit{s}}_{\mathit{i}0}^{\mathit{s}}$ = direct distance between student
**i**of school**s**and and school**s**locations **Speed**: bus speed**Timeratio**: maximum allowed ratio for students (in bus trip time/direct time to school)**cycle_time**= 60 min; allowed time for transferring each of middle school, high school, and elementary students**C**= capacity of buses**M**: a big enough number- ${\mathbf{C}}_{\mathbf{T}}:\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\text{}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}\text{}\mathrm{o}\mathrm{f}$ each student per hour
- ${\mathbf{C}}_{\mathbf{O}}:\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\text{}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}\text{}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{t}\text{}\mathrm{o}\mathrm{f}\text{}\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{o}\mathrm{l}\text{}\mathrm{b}\mathrm{u}\mathrm{s}\text{}\mathrm{p}\mathrm{e}\mathrm{r}\text{}\mathrm{k}\mathrm{i}\mathrm{l}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}$

**Variables:**

**Objective function**

**Constraints**

## 4. Algorithm

Algorithm 1 The developed BRA for origin assignment |

Step 0: Initialization:Set s = 1 (school index), desired_agents = J, i = 1(index of student), selected_students = None, remained_ students = All students, Step 1: Calculate sum of distance of each school s student’s home from others Distance (i) = sum of distance of student i home from other students’ home Step 2: IF i < I, THEN i = i+1 go to step 1, otherwise go to step 3 END IF Step 3: Find agents for school s: Step 3.1: IF number of students in remained_ students set < desired_agents, THEN go to step 3.2, otherwise go to step 4 END IF Step 3.2: Find student i, from remained_students set in which his home is closest to others (find the minimum of distance) Step 3.3: Add i to selected_students set Step 3.4: Remove i from remained_students set and go step 3 Step 4: IF s < S, THEN set s = s + 1 and go to step 1, otherwise go to step 5 END Step 5: Set s = 1, a = 1 and go to step 6 Step 6: Find origin for school s: Step 6.0: Initialization: remained_origins = set of available origins, selected_origins set for school s = Null Step 6.1 : IF s <= S, THEN go to step 6.2, otherwise go to step 7 END IF Step 6.2: IF a <= desired_agents, THEN go to step 6.3, otherwise go to step 6.1 END IF Step 6.3: Find origin o, from remained_origins set which is closest to the agent a (student) home. Step 6.4: Remove o from remained_origins set and go step 6.5 Step 6.5: Add o to selected_origins set for school s Step 6.6: set a = a+1 and go to step 6.2 Step 6.7: set s = s+1 and go to step 6.1 Step 7: End |

_{O}is the unit operating cost of each vehicle per kilometer and C

_{T}is the time value of each passenger per hour. Basically, the SA algorithm improves the solution by using two transformation methods, move and replace, to explore the solution space. The move transformation explores groups of passengers that have the shortest distances closest to each other, including their destinations. Then, it selects a random route and allocates the random passengers on the routes according to the constraints. The replace highest average transformation is based on the average distance of every group of passengers; therefore, the algorithm selects random routes and allocates selected passengers in the route in order to minimize the cost. This permutable process (cooling schedule) will be ended when the temperature reaches below 0.001 and the final solution does not change in iterations. Finally, the best feasible solution found during the total iterations is presented as the final solution proposed by the algorithm.

Algorithm 2 The developed SA algorithm to solve the proposed SBRP |

Step 0: Initialization:Set s = 1 (school index), Best Cost = positive infinite, T = T _{0}, alpha = 0.99, J = number of buses assigned to school s, Step 1: Create random solution Considering the length of trip (number of students of school s + buses of school s (J) − 1) set x as a random solution Step 2: Find optimal solution: IF It1 < It1max, THEN go to step 3, otherwise go to step 5 END IF Step 3: IF It2 < It2max, THEN go to step 3.1, otherwise go to step 4 END IF Step 3.1: Creating neighborhood: set x _{new} = a neighborhood of x Step 3.2: IF best cost for x < best cost for x_{new}, THEN set x = x _{new} and go to step 4.5, otherwise go to step 4.3 END IF Step 3.3: p = exp-(cost x _{new} − cost x)/T*Cost x Step 3.4: Accept x = x _{new} by p -probability and reject- and x = x_{new} by (1 − p) and go to step 4.5 Step 3.5: Cost calculation for x_{new} Step 3.6: IF best cost for x_{new} > best cost, THEN set bestsol = x _{new} END IF Step 3.7: Reducing the temperature: set T = alpha*T0 (0 < alpha < 1) Step 3.8: set It2 = It2 + 1 and go to step 3 Step 4: Set It1 = It1 + 1 and go to step 2 Step 5: IF bestsol is feasible, THEN go to step 7, otherwise go to step 6 END IF Step 6: Show “The problem is not feasible; more vehicles is needed” Step 7: IF s < S, THEN set s = s + 1 and go to step 1, otherwise go to step 8 END IF Step 8: Show results Step 9: END |

## 5. Example

## 6. Results

## 7. Further Discussions

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Study | Network | Problem Scope | Problem Size | Objective Function | Constraints | Approach | |
---|---|---|---|---|---|---|---|

Schools | Students | ||||||

Hargroves and Demetsky [30] | R | S | 21 | 8537 | F, TBD | C, MRT | Computer-based analytical |

Russell and Morrel [31] | R | S | 140 | TBD | C, MRT | Clarke–Wright algorithm | |

Chen, Kallsen [32] | R | S | 6 | 2413 | F, TBD | C, MRT | Computer-based analytical |

Braca, Bramel [25] | R | S | 73 | 838 | F | C, MRT, TW | Location-based heuristic |

Spada, Bierlaire [33] | R | S | 12 | 274 | STL | C, TW | Simulated annealing and tabu search |

Fügenschuh [34] | R | S | 348 | F, TBD | TW | Heuristics | |

Park, Tae [35] | R | M | 100 | 32,048 | F | C, MRT, TW | Mixed load algorithms |

Faraj, Sarubbi [36] | R | S | 23 | 944 | TBD | C, MRT, TW | GRASP-like heuristic |

Campbell, North [37] | R | S | 100 | 2000 | TBD | C, MRT, TW | Three-phase heuristic |

Bögl, Doerner [38] | H | S | 8 | 500 | TBD, SWD | C, TW | Heuristics |

Yao, Cao [39] | H | S | 2 | 1088 | TBD | C | Ant colony optimization |

de Souza Lima, Pereira [40] | R | S | 20 | 500 | TBD | C | Multi-objective iterated local search |

Caceres, Batta [41] | R | M | 118 | F, TBD | C, MRT, TW | Column-generation based | |

Miranda, de Camargo [42] | R | M | 65 | 2774 | F, TBD | C, MRT, MWT, TW | Iterated local search |

Mokhtari and Ghezavati [43] | H | S | 50 | 5906 | F | C, MRT, TW | Hybrid multi-objective ant colony optimization |

Babaei and Rajabi-Bahaabadi [44] | H | S | 100 | 434 | TBD, TSD | C, TW | Hybrid route decomposition heuristic and ant colony optimization |

Current study | H | IM | 9 | 720 | TBD, TSD | C, MRT, TW | Simulated annealing |

Routing Status | School | Time Window | Total Student Traveling Time (h) | Total Bus Traveling Distance (km) | Total Students Traveling Cost (USD) | Total Bus Operating Cost (USD) | Total Cost |
---|---|---|---|---|---|---|---|

(USD) | |||||||

Different routing for morning and afternoon (integrated framework) | HS | M | 12.74 | 62.61 | 127.35 | 187.82 | 1815.96 |

MS | M | 12.29 | 61.6 | 122.86 | 184.81 | ||

ES | M | 11.18 | 60.81 | 111.82 | 182.44 | ||

HS | A | 12.84 | 59.57 | 128.4 | 178.71 | ||

MS | A | 11.7 | 63.61 | 117.01 | 190.83 | ||

ES | A | 11.96 | 54.76 | 119.63 | 164.27 | ||

Identically reversed same routing (integrated framework) | HS | M | 14.61 | 60.22 | 146.1 | 180.65 | 1979.88 |

MS | M | 13.98 | 61.5 | 139.82 | 184.51 | ||

ES | M | 12.42 | 56.67 | 124.16 | 170.02 | ||

HS | A | 15.71 | 60.22 | 157.09 | 180.65 | ||

MS | A | 17.3 | 61.5 | 172.95 | 184.51 | ||

ES | A | 16.94 | 56.67 | 169.41 | 170.02 | ||

Different routing for morning and afternoon (separated framework) | HS | M | 14.59 | 64.11 | 145.90 | 192.33 | 2001.64 |

MS | M | 13.91 | 63.21 | 139.10 | 189.63 | ||

ES | M | 12.34 | 62.48 | 123.40 | 187.44 | ||

HS | A | 15.78 | 60.54 | 157.80 | 181.62 | ||

MS | A | 16.24 | 63.14 | 162.40 | 189.42 | ||

ES | A | 17.03 | 54.1 | 170.30 | 162.30 | ||

Identically reversed same routing (separated framework) | HS | M | 15.15 | 64.61 | 151.5 | 193.83 | 2072.05 |

MS | M | 14.56 | 63.88 | 145.56 | 191.65 | ||

ES | M | 12.9 | 63.37 | 129 | 190.1 | ||

HS | A | 16.34 | 61.83 | 163.37 | 185.5 | ||

MS | A | 17.9 | 65.77 | 179.01 | 197.32 | ||

ES | A | 17.55 | 56.57 | 175.51 | 169.7 |

Total Cost (USD) | Separated Models | Integrated Framework |
---|---|---|

The same routing for morning and afternoon (identically reversed) | 2072.05 | 1979.88 (Saves about %4.5) |

Different routing for morning and afternoon | 2001.64 (Saves about %3.5) | 1815.96 (Saves about %12.5) |

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**MDPI and ACS Style**

Nickkar, A.; Lee, Y.-J.
Optimal Integrated Single-Framework Algorithm for the Multi-Level School Bus Network Problem. *Algorithms* **2023**, *16*, 528.
https://doi.org/10.3390/a16110528

**AMA Style**

Nickkar A, Lee Y-J.
Optimal Integrated Single-Framework Algorithm for the Multi-Level School Bus Network Problem. *Algorithms*. 2023; 16(11):528.
https://doi.org/10.3390/a16110528

**Chicago/Turabian Style**

Nickkar, Amirreza, and Young-Jae Lee.
2023. "Optimal Integrated Single-Framework Algorithm for the Multi-Level School Bus Network Problem" *Algorithms* 16, no. 11: 528.
https://doi.org/10.3390/a16110528