# Testing New Methods for Boarding a Partially Occupied Airplane Using Apron Buses

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## Abstract

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## 1. Introduction

## 2. Literature Review

#### 2.1. Summary of One-Door Boarding Methods

#### 2.2. Summary of Two-Door Boarding Methods

## 3. Passenger Seat Assignment, Movement, and Luggage Assumptions

#### 3.1. Passenger Seat Assignment

- $pRow$ and $pColumn$ are passenger preference indicators, having values between 1 and 10.
- $iRow,iColumn$, and $iAgglomeration$ are passenger importance indicators, having values between 1 and 10.
- $AgglomerationState$ is a variable taking 0 or 10 as values.

#### 3.2. Passenger Movement Assumptions

#### 3.3. Luggage Assumptions

- Tstore is the time to store the luggage.
- NbinLarge is the number of large bags in the bin prior to the passenger’s arrival.
- NbinSmall is the number of small bags in the bin prior to the passenger’s arrival.
- NpassengerLarge is the number of large bags carried by the passenger.
- NpassengerSmall is the number of small bags carried by the passenger.
- Trow is the time for a passenger to walk from one row to the next (when not delayed by another passenger in front of them).

## 4. Numerical Results

- Case 1: random seat assignment (results discussed in Section 4.1).
- Case 2: random with seat assignment preferences (results discussed in Section 4.2).

#### 4.1. Random Passenger Seating Assignment

**bold**font, and the best (well) performing methods are dark (light) shaded. Observe that reverse pyramid–A, hybrid–A, and hybrid–B had the shortest average boarding times. Comparing the results gathered through reverse pyramid–A with the back-to-front approach, the average improvement in boarding time was 22.66%; when comparing it to the random boarding method, a 29.55% time improvement was found. Translated into minutes, the use of the reverse pyramid–A boarding method can reduce the boarding time, on average, by 1 minute and 37 seconds for an 80% passenger occupancy level, random seating assignment, and an S4 luggage situation when compared to the back-to-front boarding method.

#### 4.2. Preferential Passengers Seat Assignments

#### 4.3. Comparing the Methods’ Performances in the Cases of Full and Partial Flights

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Airplane Boarding Methods when Two Apron Buses and Two Airplane Doors are Used

## Appendix B. Passenger Seating Preferences for Rows and Columns

## References

- Delcea, C.; Cotfas, L.-A.; Chiriță, N.; Nica, I. A Two-Door Airplane Boarding Approach When Using Apron Buses. Sustainability
**2018**, 10, 3619. [Google Scholar] [CrossRef] - Schiphol Traffic and Transport Figures per Month. Available online: https://www.schiphol.nl/en/schiphol-group/page/transport-and-traffic-statistics/ (accessed on 13 July 2019).
- Nyquist, D.C.; McFadden, K.L. A study of the airline boarding problem. J. Air Transp. Manag.
**2008**, 14, 197–204. [Google Scholar] [CrossRef] - Steiner, A.; Philipp, M. Speeding up the airplane boarding process by using pre-boarding areas. In Proceedings of the Swiss Transport Research Conference, Ascona, Switzerland, 9–11 September 2009. [Google Scholar]
- Eurocontrol All-Causes Delay and Cancellations to Air Transport in Europe 2017. Available online: https://www.eurocontrol.int/sites/default/files/publication/files/coda-digest-annual-2017.pdf (accessed on 10 March 2019).
- Delcea, C.; Cotfas, L.-A.; Crăciun, L.; Molanescu, A.G. Are Seat and Aisle Interferences Affecting the Overall Airplane Boarding Time? An Agent-Based Approach. Sustainability
**2018**, 10, 4217. [Google Scholar] [CrossRef] - Milne, R.J.; Delcea, C.; Cotfas, L.-A.; Salari, M. New methods for two-door airplane boarding using apron buses. J. Air Transp. Manag.
**2019**, 80, 101705. [Google Scholar] [CrossRef] - Delcea, C.; Cotfas, L.-A.; Salari, M.; Milne, R.J. Investigating the Random Seat Boarding Method without Seat Assignments with Common Boarding Practices Using an Agent-Based Modeling. Sustainability
**2018**, 10, 4623. [Google Scholar] [CrossRef] - Ferrari, P.; Nagel, K. Robustness of Efficient Passenger Boarding Strategies for Airplanes. Transp. Res. Rec. J. Transp. Res. Board
**2005**, 1915, 44–54. [Google Scholar] [CrossRef] - Wilensky, U.; Rand, W. An Introduction to Agent-Based Modeling: Modeling Natural, Social, and Engineered Complex Systems with Netlogo; The MIT Press: Cambridge, MA, USA, 2015; ISBN 978-0-262-73189-8. [Google Scholar]
- Schultz, M. Field Trial Measurements to Validate a Stochastic Aircraft Boarding Model. Aerospace
**2018**, 5, 27. [Google Scholar] [CrossRef] - Schultz, M. A metric for the real-time evaluation of the aircraft boarding progress. Transp. Res. Part C Emerg. Technol.
**2018**, 86, 467–487. [Google Scholar] [CrossRef] - Milne, R.J.; Kelly, A.R. A new method for boarding passengers onto an airplane. J. Air Transp. Manag.
**2014**, 34, 93–100. [Google Scholar] [CrossRef] [Green Version] - Jafer, S.; Mi, W. Comparative Study of Aircraft Boarding Strategies Using Cellular Discrete Event Simulation. Aerospace
**2017**, 4, 57. [Google Scholar] [CrossRef] - Bachmat, E.; Berend, D.; Sapir, L.; Skiena, S.; Stolyarov, N. Analysis of Airplane Boarding Times. Oper. Res.
**2009**, 57, 499–513. [Google Scholar] [CrossRef] [Green Version] - Qiang, S.-J.; Jia, B.; Xie, D.-F.; Gao, Z.-Y. Reducing airplane boarding time by accounting for passengers’ individual properties: A simulation based on cellular automaton. J. Air Transp. Manag.
**2014**, 40, 42–47. [Google Scholar] [CrossRef] - Notomista, G.; Selvaggio, M.; Sbrizzi, F.; Di Maio, G.; Grazioso, S.; Botsch, M. A fast airplane boarding strategy using online seat assignment based on passenger classification. J. Air Transp. Manag.
**2016**, 53, 140–149. [Google Scholar] [CrossRef] - Ren, X.; Xu, X. Experimental analyses of airplane boarding based on interference classification. J. Air Transp. Manag.
**2018**, 71, 55–63. [Google Scholar] [CrossRef] - Stonedahl, F.; Wilensky, U. Finding Forms of Flocking: Evolutionary Search in ABM Parameter-Spaces; Springer Science and Business Media LLC: Berlin/Heidelberg, Germany, 2011; Volume 6532, pp. 61–75. [Google Scholar]
- Schultz, M. Implementation and application of a stochastic aircraft boarding model. Transp. Res. Part C Emerg. Technol.
**2018**, 90, 334–349. [Google Scholar] [CrossRef] - Steffen, J.H. Optimal boarding method for airline passengers. J. Air Transp. Manag.
**2008**, 14, 146–150. [Google Scholar] [CrossRef] [Green Version] - Soolaki, M.; Mahdavi, I.; Mahdavi-Amiri, N.; Hassanzadeh, R.; Aghajani, A. A new linear programming approach and genetic algorithm for solving airline boarding problem. Appl. Math. Model.
**2012**, 36, 4060–4072. [Google Scholar] [CrossRef] - Milne, R.J.; Salari, M.; Kattan, L. Robust Optimization of Airplane Passenger Seating Assignments. Aerospace
**2018**, 5, 80. [Google Scholar] [CrossRef] - Tang, T.-Q.; Yang, S.-P.; Ou, H.; Chen, L.; Huang, H.-J. An aircraft boarding model with the group behavior and the quantity of luggage. Transp. Res. Part C Emerg. Technol.
**2018**, 93, 115–127. [Google Scholar] [CrossRef] - Van Den Briel, M.H.L.; Villalobos, J.R.; Hogg, G.L.; Lindemann, T.; Mulé, A.V. America West Airlines Develops Efficient Boarding Strategies. Interfaces
**2005**, 35, 191–201. [Google Scholar] [CrossRef] - Milne, R.J.; Salari, M. Optimization of assigning passengers to seats on airplanes based on their carry-on luggage. J. Air Transp. Manag.
**2016**, 54, 104–110. [Google Scholar] [CrossRef] - Steffen, J.H. A statistical mechanics model for free-for-all airplane passenger boarding. Am. J. Phys.
**2008**, 76, 1114–1119. [Google Scholar] [CrossRef] [Green Version] - Steffen, J.H.; Hotchkiss, J. Experimental test of airplane boarding methods. J. Air Transp. Manag.
**2012**, 18, 64–67. [Google Scholar] [CrossRef] [Green Version] - Hutter, L.; Jaehn, F.; Neumann, S. Influencing Factors on Airplane Boarding Times; Omega: Stamford, CT, USA, 2018. [Google Scholar]
- Kierzkowski, A.; Kisiel, T. The Human Factor in the Passenger Boarding Process at the Airport. Procedia Eng.
**2017**, 187, 348–355. [Google Scholar] [CrossRef] - Van Landeghem, H.; Beuselinck, A. Reducing passenger boarding time in airplanes: A simulation based approach. Eur. J. Oper. Res.
**2002**, 142, 294–308. [Google Scholar] [CrossRef] - Qiang, S.; Jia, B.; Huang, Q. Evaluation of Airplane Boarding/Deboarding Strategies: A Surrogate Experimental Test. Symmetry
**2017**, 9, 222. [Google Scholar] [CrossRef] - Delcea, C.; Cotfas, L.-A.; Paun, R. Agent-Based Evaluation of the Airplane Boarding Strategies’ Efficiency and Sustainability. Sustainability
**2018**, 10, 1879. [Google Scholar] [CrossRef] - Schultz, M.; Kunze, T.; Fricke, H. Boarding on the Critical Path of the Turnaround. In Proceedings of the 10th USA/Europe Air Traffic Management Research and Development Seminar, Chicago, IL, USA, 10–13 June 2013; pp. 1–10. [Google Scholar]
- Schultz, M.; Schulz, C.; Fricke, H. Efficiency of Aircraft Boarding Procedures. In Proceedings of the 3rd International Conference on Research in Airport Transportation, Madrid, Spain, 1–3 October 2008; Volume 371–391. [Google Scholar]
- Jaehn, F.; Neumann, S. Airplane boarding. Eur. J. Oper. Res.
**2015**, 244, 339–359. [Google Scholar] [CrossRef] - Marelli, S.; Mattocks, G.; Merry, R. The role of computer simulation in reducing airplane turnaround time. Boeing Aero Mag.
**1998**, 1. [Google Scholar] - Gao, M.; Zhou, L.; Chen, Y. An Alternative Approach for High Speed Railway Carrying Capacity Calculation Based on Multiagent Simulation. Discret. Dyn. Nat. Soc.
**2016**, 2016, 4278073. [Google Scholar] [CrossRef] - Vo, T.T.A.; Van Der Waerden, P.; Wets, G. Micro-simulation of Car Drivers’ Movements at Parking Lots. Procedia Eng.
**2016**, 142, 100–107. [Google Scholar] [CrossRef] - Shqair, M.; Altarazi, S.; Al-Shihabi, S. A statistical study employing agent-based modeling to estimate the effects of different warehouse parameters on the distance traveled in warehouses. Simul. Model. Pr. Theory
**2014**, 49, 122–135. [Google Scholar] [CrossRef] - Faroqi, H.; Mesgari, M.-S. Agent-Based Crowd Simulation Considering Emotion Contagion For Emergency Evacuation Problem. ISPRS Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci.
**2015**, XL-1-W5, 193–196. [Google Scholar] [CrossRef] - Liu, R.; Jiang, D.; Shi, L. Agent-based simulation of alternative classroom evacuation scenarios. Front. Arch. Res.
**2016**, 5, 111–125. [Google Scholar] [CrossRef] [Green Version] - Gutierrez-Milla, A.; Borges, F.; Suppi, R.; Luque, E. Individual-oriented Model Crowd Evacuations Distributed Simulation. Procedia Comput. Sci.
**2014**, 29, 1600–1609. [Google Scholar] [CrossRef] [Green Version] - Wang, H.; Mostafizi, A.; Cramer, L.A.; Cox, D.; Park, H. An agent-based model of a multimodal near-field tsunami evacuation: Decision-making and life safety. Transp. Res. Part C Emerg. Technol.
**2016**, 64, 86–100. [Google Scholar] [CrossRef] - Delcea, C.; Cotfas, L.-A. Increasing awareness in classroom evacuation situations using agent-based modeling. Phys. A Stat. Mech. Appl.
**2019**, 523, 1400–1418. [Google Scholar] [CrossRef] - Delcea, C.; Cotfas, L.-A.; Trică, C.L.; Crăciun, L.; Molanescu, A.G. Modeling the Consumers Opinion Influence in Online Social Media in the Case of Eco-friendly Products. Sustainability
**2019**, 11, 1796. [Google Scholar] [CrossRef] - Wooldridge, M.; Jennings, N.R. Intelligent agents: Theory and practice. Knowl. Eng. Rev.
**1995**, 10, 115–152. [Google Scholar] [CrossRef] - Getchell, A. Agent-Based Modeling. Available online: http://rgdoi.net/10.13140/RG.2.1.2880.8803,2015 (accessed on 10 March 2019).
- Delcea, C.; Bradea, I.A. Economic Cybernetics. An Equation-Based Modeling and Agent-Based Modeling Approach; Editura Universitara: Bucharest, Romania, 2017; ISBN 978-606-28-0629-3. [Google Scholar]
- Schultz, M. Fast Aircraft Turnaround Enabled by Reliable Passenger Boarding. Aerospace
**2018**, 5, 8. [Google Scholar] [CrossRef] - Bazargan, M. A linear programming approach for aircraft boarding strategy. Eur. J. Oper. Res.
**2007**, 183, 394–411. [Google Scholar] [CrossRef] - Audenaert, J.; Verbeeck, K.; Berghe, G. Multi-agent based simulation for boarding. In Proceedings of the 21st Benelux Conference on Artificial Intelligence, Eindhoven, The Netherlands, 29–30 October 2009; pp. 3–10. [Google Scholar]

**Figure 10.**Example of seating for an 80% passenger occupancy level with preferential seat assignments.

Classification | Boarding Method | Summary of the Method |
---|---|---|

Random | Random without assigned seats | Passengers board the airplane through a first-come first-serve basis. The passengers do not have the option to select their seats prior to boarding, nor does the airline company assign them seats. This method is also called free-for-all boarding. |

Random with assigned seats | The passengers have a seat assigned on their ticket. The boarding process (sequence in which passengers board) is made in a random manner. Once inside the aircraft, each passenger proceeds to the seat indicated on the boarding pass. | |

By group | Outside-in (also known as WilMA) | The passengers are divided into three main groups, each group being individually called for boarding, starting with the seats near the window, continuing with the middle seats, and ending with the seats near the aisle. Within each group, the passengers board randomly. |

Reverse pyramid | The groups are boarded using a diagonal scheme, starting with passengers near the window in the rear rows of the airplane and ending with those having aisle seats in the front rows. | |

Back-to-front | The passengers are divided into five or more groups based on the rows in which they have their seats. The first group to board is formed by passengers having seats in the rear of the airplane, while the last group to board contains passengers with seats in the front of the airplane. | |

By seat | Back to front by seating order | As with all “by seat” strategies, the passengers are called for boarding one-by-one using a given scheme. In this case, the passenger sitting in the very last row of the airplane in the column identified by letter A is called first, followed by the passenger with letter F, B, E, C, and D. Then the last-but-one row of passengers is called one-by-one. The last person called to board has the seat in the first row, with letter D. |

Steffen method | The “by seat” approach applies in this case too. The first passenger called for boarding has the seat in the last row, near the window. The second passenger to board has a seat near the window but two rows in front of the first-called passenger. The scheme continues every other row until reaching the front of the airplane, then continues in the same way on the other side of the airplane near the window. After that, the “missed” seats near the first window are filed-in following the same scheme, then the free seats near the other window follow. The scheme continues in the same way for passengers to be seated in the middle and aisle seats. |

Classification | Boarding Method | Summary of the Method |
---|---|---|

By group | Back-to-front | The airplane is divided into four equal groups starting from the rear of the airplane. The passengers having the seats in the rows located in the middle, groups 2 and 3, are boarded in the first apron bus, while the passengers with seats in the front and in the rear of the airplane board in the second apron bus (Figure A1). |

Spread across rows | The method was inspired by the WilMA method and applies some of the WilMa concepts to the case in which only two boarding groups can be formed. For the first apron bus, all the passengers having the seats near the window are assigned, along with half of the passengers having middle seats, while the second apron bus carries all the passengers having the seats near the aisle and the remainder of the passengers with the middle seats. The complete scheme for an Airbus A320 is presented in Figure 4. Note that in each row there is an equal number of passengers to be boarded in the first and second apron bus. | |

Half-spread across rows | The method is inspired by the spread across rows and back-to-front methods. As a result, the passengers with window seats are assigned to the first bus, along with most of the passengers located in the two middle groups formed in the back-to-front method case. The remainder of the passengers are allocated to the second apron bus (Figure 5). | |

Variations in reverse pyramid (from A to F) | The variations in the reverse pyramid combine concepts from the rules of the WilMA, back-to-front, and reverse pyramid methods applied in the case of the use of two apron buses. The first variation, reverse pyramid–A is presented in Figure 6. The first apron bus includes all the passengers with seats near the window and all the passengers with middle seats located in the middle part of the airplane, which are groups 2 and 3 from the back-to-front method. As we move further to the other five variations in reverse pyramid (reverse pyramid–B to reverse pyramid–F), the scheme for assigning passengers to the buses is modified for each iteration, where some of the seats located near the window in the front and in the rear of the airplane will be no longer allocated to the first bus, but rather will be assigned to the second bus. In compensation, more seats adjacent to the aisle in the middle part of the airplane will be assigned to the first bus. The schemes for reverse pyramid–B to reverse pyramid–F are given in Figure A2, Figure A3, Figure A4, Figure A5 and Figure A6. | |

Variations in hybrid methods (from A to E) | The variations in the hybrid methods leverage concepts from the reverse pyramid–A and spread across rows method. The first variation, hybrid–A is slightly different from reverse pyramid–A (Figure 7), while hybrid–B to hybrid–E (Figure A7, Figure A8, Figure A9 and Figure A10) in turn provide an increasing number of rows with three passengers assigned to each apron bus. |

Types of Seat Interferences | Type 1 | Type 2 | Type 3 | Type 4 |
---|---|---|---|---|

Scheme | ||||

Average duration (seconds) | 22 | 12 | 10 | 10 |

Range (seconds) | [20,26] | [10,13] | [9,13] | [9,13] |

Situation | Percentages of Bags Carried by the Passengers | ||||
---|---|---|---|---|---|

0 Bag | 1 Small Bag | 2 Small Bags | 1 Large Bag | 1 Large and 1 Small Bag | |

S1 | 10% | 10% | 0% | 10% | 70% |

S2 | 15% | 20% | 5% | 10% | 50% |

S3 | 25% | 20% | 10% | 15% | 30% |

S4 | 35% | 25% | 10% | 15% | 15% |

S5 | 60% | 10% | 10% | 10% | 10% |

S6 | 80% | 5% | 5% | 5% | 5% |

S7 | 100% | 0% | 0% | 0% | 0% |

**Table 5.**The average boarding times (in ticks) for partial occupancy levels, random seat assignments, and the S4 luggage situation.

Boarding Method | Luggage Situation: S4 | |||||
---|---|---|---|---|---|---|

Occupancy Level: | Average Boarding Time | Boarding Time Improvement When Compared to Random | ||||

60% | 70% | 80% | 90% | |||

Benchmark1: Random | 203 | 240 | 273 | 316 | 258.00 | |

Benchmark2: Back-to-front | 177 | 218 | 252 | 293 | 235.00 | |

Spread across rows | 160 | 184 | 208 | 233 | 196.25 | 23.93% |

Half spread across rows | 155 | 179 | 201 | 228 | 190.75 | 26.07% |

Reverse pyramid–A | 149 | 172 | 192 | 214 | 181.75 | 29.55% |

Reverse pyramid–B | 151 | 180 | 194 | 215 | 185.00 | 28.29% |

Reverse pyramid–C | 159 | 185 | 195 | 216 | 188.75 | 26.84% |

Reverse pyramid–D | 160 | 189 | 195 | 217 | 190.25 | 26.26% |

Reverse pyramid–E | 162 | 190 | 199 | 221 | 193.00 | 25.19% |

Reverse pyramid–F | 164 | 199 | 232 | 267 | 215.50 | 16.47% |

Hybrid–A | 149 | 171 | 193 | 213 | 181.50 | 29.65% |

Hybrid–B | 149 | 171 | 207 | 217 | 186.00 | 27.91% |

Hybrid–C | 152 | 173 | 213 | 217 | 188.75 | 26.84% |

Hybrid–D | 151 | 173 | 218 | 223 | 191.25 | 25.87% |

Hybrid–E | 152 | 174 | 229 | 231 | 196.50 | 23.84% |

**Table 6.**The average boarding times (in ticks) for 80% occupancy level, random seat assignments, and S1-S7 luggage situations.

Boarding Method | Occupancy Level: 80% | ||||||||
---|---|---|---|---|---|---|---|---|---|

Luggage Situations: | Average Boarding Time | Boarding Time Improvement When Compared to Random | |||||||

S1 | S2 | S3 | S4 | S5 | S6 | S7 | |||

Benchmark1: Random | 339 | 315 | 297 | 273 | 262 | 239 | 215 | 277.14 | |

Benchmark2: Back-to-front | 314 | 293 | 269 | 252 | 233 | 220 | 190 | 253.00 | |

Spread across rows | 270 | 249 | 228 | 208 | 190 | 171 | 145 | 208.71 | 24.69% |

Half spread across rows | 258 | 240 | 219 | 201 | 185 | 168 | 141 | 201.71 | 27.22% |

Reverse pyramid–A | 251 | 229 | 212 | 192 | 177 | 161 | 132 | 193.43 | 30.21% |

Reverse pyramid–B | 260 | 241 | 219 | 194 | 190 | 174 | 145 | 203.29 | 26.65% |

Reverse pyramid–C | 263 | 246 | 232 | 195 | 195 | 180 | 156 | 209.57 | 24.38% |

Reverse pyramid–D | 271 | 253 | 235 | 195 | 209 | 188 | 157 | 215.43 | 22.27% |

Reverse pyramid–E | 283 | 260 | 244 | 199 | 212 | 192 | 168 | 222.57 | 19.69% |

Reverse pyramid–F | 291 | 267 | 247 | 232 | 215 | 200 | 176 | 232.57 | 16.08% |

Hybrid–A | 249 | 233 | 210 | 193 | 179 | 159 | 132 | 193.57 | 30.15% |

Hybrid–B | 251 | 232 | 213 | 207 | 178 | 159 | 133 | 196.14 | 29.23% |

Hybrid–C | 252 | 232 | 213 | 213 | 179 | 160 | 134 | 197.57 | 28.71% |

Hybrid–D | 252 | 234 | 214 | 218 | 178 | 160 | 132 | 198.29 | 28.45% |

Hybrid–E | 254 | 234 | 213 | 229 | 182 | 162 | 132 | 200.86 | 27.53% |

**Table 7.**The average boarding times (in ticks) for partial occupancy levels, preferential seat assignments, and S4 luggage situation.

Boarding Method | Luggage Situation: S4 | |||||
---|---|---|---|---|---|---|

Occupancy Level: | Average Boarding Time | Boarding Time Improvement When Compared to Random | ||||

60% | 70% | 80% | 90% | |||

Benchmark1: Random | 199 | 248 | 286 | 325 | 264.50 | |

Benchmark2: Back-to-front | 174 | 219 | 264 | 306 | 240.75 | |

Spread across rows | 154 | 184 | 209 | 236 | 195.75 | 25.99% |

Half spread across rows | 145 | 178 | 203 | 232 | 189.50 | 28.36% |

Reverse pyramid–A | 142 | 168 | 197 | 218 | 181.25 | 31.47% |

Reverse pyramid–B | 145 | 177 | 206 | 229 | 189.25 | 28.45% |

Reverse pyramid–C | 150 | 185 | 217 | 241 | 198.25 | 25.05% |

Reverse pyramid–D | 151 | 189 | 222 | 256 | 204.50 | 22.68% |

Reverse pyramid–E | 156 | 194 | 234 | 265 | 212.25 | 19.75% |

Reverse pyramid–F | 161 | 202 | 241 | 274 | 219.50 | 17.01% |

Hybrid–A | 142 | 169 | 196 | 220 | 181.75 | 31.29% |

Hybrid–B | 143 | 172 | 198 | 220 | 183.25 | 30.72% |

Hybrid–C | 144 | 170 | 196 | 220 | 182.50 | 31.00% |

Hybrid–D | 144 | 172 | 199 | 221 | 184.00 | 30.43% |

Hybrid–E | 146 | 174 | 199 | 223 | 185.50 | 29.87% |

**Table 8.**The average boarding times (in ticks) for an 80% occupancy level, preferential seat assignments, and S1–S7 luggage situations.

Boarding Method | Occupancy Level: 80% | ||||||||
---|---|---|---|---|---|---|---|---|---|

Luggage Situations: | Average Boarding Time | Boarding Time Improvement When Compared to Random | |||||||

S1 | S2 | S3 | S4 | S5 | S6 | S7 | |||

Benchmark1: Random | 362 | 332 | 306 | 286 | 268 | 243 | 220 | 288.14 | |

Benchmark2: Back-to-front | 334 | 311 | 283 | 264 | 246 | 228 | 204 | 267.14 | |

Spread across rows | 276 | 252 | 232 | 209 | 191 | 173 | 147 | 211.43 | 26.62% |

Half spread across rows | 268 | 246 | 224 | 203 | 189 | 170 | 141 | 205.86 | 28.56% |

Reverse pyramid–A | 257 | 237 | 215 | 197 | 177 | 161 | 134 | 196.86 | 31.68% |

Reverse pyramid–B | 262 | 244 | 223 | 206 | 186 | 172 | 143 | 205.14 | 28.81% |

Reverse pyramid–C | 272 | 252 | 233 | 217 | 196 | 178 | 152 | 214.29 | 25.63% |

Reverse pyramid–D | 283 | 262 | 239 | 222 | 212 | 190 | 160 | 224.00 | 22.26% |

Reverse pyramid–E | 295 | 268 | 249 | 234 | 215 | 196 | 171 | 232.57 | 19.29% |

Reverse pyramid–F | 303 | 285 | 260 | 241 | 228 | 208 | 182 | 243.86 | 15.37% |

Hybrid–A | 255 | 239 | 215 | 196 | 179 | 160 | 134 | 196.86 | 31.68% |

Hybrid–B | 254 | 237 | 215 | 198 | 183 | 160 | 134 | 197.29 | 31.53% |

Hybrid–C | 263 | 234 | 216 | 196 | 183 | 161 | 133 | 198.00 | 31.28% |

Hybrid–D | 262 | 237 | 217 | 199 | 183 | 160 | 135 | 199.00 | 30.94% |

Hybrid–E | 262 | 241 | 219 | 199 | 183 | 161 | 134 | 199.86 | 30.64% |

**Table 9.**Boarding time improvements when compared to the results of the back-to-front method (in %).

Boarding Method | Luggage Situation: S4 | ||
---|---|---|---|

Occupancy Level: | |||

80% | 80% | 100% | |

Passengers Seat Assignment: | |||

Random | Preferences | Random/Preferences | |

Spread across rows | 20.83% | 22.36% | 22.45% |

Half spread across rows | 23.11% | 23.17% | 23.62% |

Reverse pyramid–A | 25.38% | 28.05% | 27.11% |

Reverse pyramid–B | 21.97% | 24.39% | 20.99% |

Reverse pyramid–C | 17.80% | 20.33% | 18.08% |

Reverse pyramid–D | 15.91% | 13.82% | 14.58% |

Reverse pyramid–E | 11.36% | 12.60% | 11.08% |

Reverse pyramid–F | 8.71% | 7.32% | 7.87% |

Hybrid–A | 25.76% | 27.24% | 27.41% |

Hybrid–B | 25.00% | 25.61% | 27.41% |

Hybrid–C | 25.76% | 25.61% | 26.82% |

Hybrid–D | 24.62% | 25.61% | 26.82% |

Hybrid–E | 24.62% | 25.61% | 26.24% |

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## Share and Cite

**MDPI and ACS Style**

Cotfas, L.-A.; Delcea, C.; Milne, R.J.; Salari, M.; Crăciun, L.; Molănescu, A.G.
Testing New Methods for Boarding a Partially Occupied Airplane Using Apron Buses. *Symmetry* **2019**, *11*, 1044.
https://doi.org/10.3390/sym11081044

**AMA Style**

Cotfas L-A, Delcea C, Milne RJ, Salari M, Crăciun L, Molănescu AG.
Testing New Methods for Boarding a Partially Occupied Airplane Using Apron Buses. *Symmetry*. 2019; 11(8):1044.
https://doi.org/10.3390/sym11081044

**Chicago/Turabian Style**

Cotfas, Liviu-Adrian, Camelia Delcea, R. John Milne, Mostafa Salari, Liliana Crăciun, and Anca Gabriela Molănescu.
2019. "Testing New Methods for Boarding a Partially Occupied Airplane Using Apron Buses" *Symmetry* 11, no. 8: 1044.
https://doi.org/10.3390/sym11081044