# Revenue Management in Airlines and External Factors Affecting Decisions: The Harmonic Oscillator Model

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## Abstract

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## 1. Introduction

## 2. The Standard Revenue Management Problem in Airlines

#### The Two Class Problem

## 3. The Quantum Harmonic Oscillator for the Single Resource Two-Class Problem

#### The Two-Dimensional Case: Single Resource Two-Class Problem

## 4. Collective Decisions in the RM Problem: Spontaneous Symmetry Breaking

## 5. Numerical Example

#### Comparison with Other Methods

## 6. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Ellipsoid illustrating the phase space covered by the Quantum Harmonic Oscillator. Without loss of generality, we can take $a={p}_{max\phantom{\rule{0.277778em}{0ex}}1}$ as the largest possible price for the first class and $b={X}_{max\phantom{\rule{0.277778em}{0ex}}1}$ as the largest possible demand for the first class. $c={p}_{max\phantom{\rule{0.277778em}{0ex}}2}$ might represent, for example, the largest possible price for the second class, with the corresponding largest demand for the second class given by ${a}_{max}$. Although we are talking about a four-dimensional ellipsoid, this figure illustrates a three-dimensional example.

**Figure 2.**Symmetry breaking pattern with the x-axis representing the demand and the y-axis representing the price. The red line corresponds to the standard situation where both ${\omega}^{2}$ and $\lambda $ are positive. In this case, we only have one well-defined vacuum state. On the other hand, the blue curve corresponds to the case where ${\omega}^{2}<0$. This situation forces the system to develop two (or infinite) different vacuums as illustrated in the figure. The most stable state when ${\omega}^{2}<0$ corresponds to the one where the symmetry is spontaneously broken.

**Figure 3.**Changes on the phase space due to the presence of the quartic-order term in the Hamiltonian (20). The x-axis represents the demand and the y-axis represents the price. Note that the available phase space looks bigger when the quartic term appears. However, due to spontaneous symmetry breaking, the phase space is not fully available and it collapses to a line.

**Table 1.**Special values for the prices for Hong Kong Express. The itinerary corresponds to the trip Hong Kong–Tokyo. The period under analysis was 15 February 2024–5 April 2024. All the prices during this period are equal to HKD 588, except for most of the dates appearing in the table.

Key Dates P | Price HKE (HKD) | Day of the Week |
---|---|---|

16 February | 608 | Friday |

23 February | 708 | Friday |

24 February | 1508 | Saturday |

25 February | 2888 | Sunday |

1 March | 608 | Friday |

3 March | 658 | Sunday |

4 March | 658 | Monday |

5 March | 658 | Tuesday |

8 March | 608 | Friday |

9 March | 652 | Saturday |

10 March | 588 | Sunday |

14 March | 658 | Thursday |

15 March | 708 | Friday |

20 March | 728 | Wednesday |

21 March | 1068 | Thursday |

22 March | 938 | Friday |

23 March | 918 | Saturday |

24 March | 918 | Sunday |

25 March | 788 | Monday |

26 March | 1268 | Tuesday |

27 March | 1488 | Wednesday |

28 March | 2018 | Thursday |

29 March | 1588 | Friday |

30 March | 1318 | Saturday |

31 March st | 818 | Sunday |

5 April | 608 | Friday |

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**MDPI and ACS Style**

Arraut, I.; Rosado, W.; Leong, V.
Revenue Management in Airlines and External Factors Affecting Decisions: The Harmonic Oscillator Model. *Mathematics* **2024**, *12*, 847.
https://doi.org/10.3390/math12060847

**AMA Style**

Arraut I, Rosado W, Leong V.
Revenue Management in Airlines and External Factors Affecting Decisions: The Harmonic Oscillator Model. *Mathematics*. 2024; 12(6):847.
https://doi.org/10.3390/math12060847

**Chicago/Turabian Style**

Arraut, Ivan, Wilson Rosado, and Victor Leong.
2024. "Revenue Management in Airlines and External Factors Affecting Decisions: The Harmonic Oscillator Model" *Mathematics* 12, no. 6: 847.
https://doi.org/10.3390/math12060847