# Dynamic Model of Contingency Flight Crew Planning Extending to Crew Formation

^{*}

## Abstract

**:**

## 1. Introduction—Motivation for Solution

- the crew member’s total hours of service did not exceed 190 h over 28 consecutive days,
- the crew member’s total hours of service did not exceed 60 h over 7 consecutive days,
- the total flying time on which the crew member is deployed does not exceed 900 h in a calendar year,
- the total flying time on which the crew member is deployed does not exceed 100 h over 28 consecutive days.
- The regulations stipulate that the base daily flight duty time of a crew member is 13 h. In staffing pilots for duty, the regulations recommend that the distribution of flight performance over time should be as even as possible.

- Using up of the allowed daily work hours,
- Health complications.

- An emergency arising at base (the airline’s home airport),
- An emergency arising away from base.

- Deploying a stand-by crew,
- Deploying a crew originally scheduled for another flight,
- Deploying pilots fulfilling other than their direct flight obligations at the airport (on duty),
- Deploying pilots currently on personal leave.

- Transport of the reserve crew to the airport of departure of the delayed flight,
- Deployment of a crew from another flight,
- Reassignment of the crew after the compulsory rest period.

- Two hours or more in the case of flights of 1500 km or less, or
- Three hours or more in the case of all flights within the European Community longer than 1500 km and all other flights between 1500 and 3500 km, or
- Four hours or more in the case of all flights not falling under (a) or (b).

## 2. Analysis of Current State of the Art

#### 2.1. Crew Scheduling

#### 2.2. More Complex Tasks Involving Crew Pairing and Crew Rostering

#### 2.3. Addressing Emergency Situations

## 3. Proposed Optimization Approach

## 4. Proposed Formulation of the Optimization Problem

#### 4.1. Problem Formulation

#### 4.2. Mathematical Model

#### Preparation of the Model

## 5. Calculation Experiments with the Proposed Model

- Crew pairing (route) plan for operating all flights (1–21).
- Re-planning of crew pairing (routes) when an emergency arises.

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Notation

Symbol | Meaning |

$I$ | Set of un-serviced flights |

$K$ | Set of crews |

${K}_{j}^{c}$ | Set of crews including captains, with whom a first officer $j\in {R}_{f}$ can form a crew |

${K}_{i}^{f}$ | Set of crews including first officers, with whom a captain $i\in {R}_{c}$ can form a crew |

$R$ | Set of pilots |

${R}_{c}$ | Subset of pilots with captain qualification |

${R}_{f}$ | Subset of pilots with first officer qualification |

$\mathit{D}$ | Three-dimensional matrix of values of penalty constants |

$\mathit{E}$ | Incidence matrix expressing permissibility of the combination of captain and first officer |

$\mathit{P}$ | Three-dimensional matrix expressing costs for transfer of crews |

$\mathit{Q}$ | Three-dimensional matrix expressing time demand for transfer of crews |

${b}_{i}$ | Delay time of flight $i\in I$, which when exceeded entitles passengers to compensation |

${d}_{ijk}$ | Element of matrix $D$, penalizing deployment of crews $k\in K$ between flights $i\in I\cup \left\{0\right\}$ and $j\in I\cup \left\{0\right\}$, which deviate from the original plan |

${e}_{ij}$ | Element of the incidence matrix expressing link between captain $i\in {R}_{c}$ and the first officer $j\in {R}_{f}$ |

$L$ | Maximum permitted daily duty hours |

$M$ | Large enough constant |

${N}_{i}$ | Estimated number of passengers on flight $i\in I$ |

${o}_{i}$ | Financial compensation of passengers for delay on flight $i\in I$ |

${p}_{ijk}$ | Element of matrix $P$expresses costs generated by non-productive transfer of crew $k\in K$ to operate flight $j\in I\cup \left\{0\right\}$ after operating flight $i\in I\cup \left\{0\right\}$ |

${q}_{ijk}$ | Element of matrix $Q$expresses time necessary for non-productive transfer of crew $k\in K$ to operate flight $j\in I\cup \left\{0\right\}$ after operating flight $i\in I\cup \left\{0\right\}$ |

${s}_{k}$ | Current elapsed on duty time for the crew $k\in K$ |

${t}_{i}$ | Planned pre-flight preparations for flight $i\in I\cup \left\{0\right\}$ |

${T}_{i}$ | Time necessary for operation of flight $i$ |

${\overline{h}}_{k}$ | Variable modeling delimiting the lower limit of the daily time in the crew’s duty $k\in K$ |

${\stackrel{\mathrm{=}}{h}}_{k}$ | Variable modeling delimiting the upper limit of the daily time in the crew’s duty $k\in K$ |

${x}_{ijk}$ | Variable deciding on the transfer of the crew $k\in K$ after operating flight $i\in I\cup \left\{0\right\}$ to operate flight $j\in I\cup \left\{0\right\}$ |

${y}_{jk}$ | Variable modeling the real value of delay in arrival of flight $i\in I$, operated by the crew $k\in K$ |

${z}_{i}$ | Variable deciding on payment of financial compensation to passengers on flight $i\in I$ |

${w}_{k}$ | Variable deciding on crew creation $k\in K$ |

$\epsilon $ | Separation constant |

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**Table 1.**Amount of financial compensation for passengers [8].

Flight Distance (km) | Minimum Length of Delay (h) | Amount of Compensation Paid to the Passenger (EUR) |
---|---|---|

$\langle 0;\left.1500\right)$ | 2 | 250 |

$\langle 1500;\left.3500\right)$ | 3 | 400 |

$\langle 3500;\left.\infty \right)$ | 4 | 600 |

**Table 2.**Incidence matrix$\text{}\mathit{E}$ representing the permissibility of creating crews with the captain and first officer.

FO/ Captain | 1 | 2 | 3 | 4 |

1 | 1 | 0 | 1 | 1 |

2 | 0 | 1 | 1 | 0 |

Captain | 1 | 1 | 1 | 2 | 2 |

First officer | 1 | 3 | 4 | 2 | 3 |

Crew | 1 | 2 | 3 | 4 | 5 |

FO | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

CAP | |||||||||||||

1 | x | x | x | 1 | 2 | 3 | x | 4 | x | x | 5 | x | |

2 | 6 | x | x | x | x | x | x | x | 7 | x | x | 8 | |

3 | 9 | 10 | x | x | x | x | 11 | x | x | 12 | x | x | |

4 | x | x | 13 | x | 14 | x | 15 | x | x | x | x | x | |

5 | 16 | 17 | x | 18 | x | x | x | x | x | x | 19 | x | |

6 | x | x | 20 | x | x | 21 | x | x | x | x | x | 22 | |

7 | x | x | x | x | x | x | x | 23 | 24 | x | x | x | |

8 | x | 25 | 26 | x | x | X | x | x | x | 27 | x | 28 | |

9 | 29 | x | x | 30 | 31 | X | x | 32 | 33 | x | x | x | |

10 | x | x | x | x | x | 34 | 35 | x | x | 36 | 37 | x |

FRA | MUC | MXP | |||
---|---|---|---|---|---|

Captains | FO | Captains | FO | Captains | FO |

1,2,3,4,5 | 1,2,3,10,11,12 | 6,7,8,10 | 4,6,7,8,9 | 9 | 5 |

$i$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

${t}_{i}\text{}\left[\mathrm{min}\right]$ | 510 | 510 | 535 | 640 | 640 | 655 | 655 | 740 | 785 | 810 |

${T}_{i}\text{}\left[\mathrm{min}\right]$ | 90 | 70 | 70 | 80 | 90 | 105 | 120 | 180 | 75 | 115 |

$i$ | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |

${t}_{i}\text{}\left[\mathrm{min}\right]$ | 825 | 880 | 930 | 970 | 1000 | 1020 | 1045 | 1105 | 1110 | 1250 | 1255 |

${T}_{i}\text{}\left[\mathrm{min}\right]$ | 230 | 120 | 135 | 175 | 70 | 85 | 125 | 75 | 130 | 120 | 70 |

$i$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

${N}_{i}$ | 110 | 80 | 85 | 90 | 110 | 105 | 75 | 70 | 90 | 110 |

${o}_{i}$ | 250 | 250 | 250 | 250 | 250 | 250 | 400 | 400 | 250 | 250 |

${b}_{i}$ | 120 | 120 | 120 | 120 | 120 | 120 | 180 | 180 | 120 | 120 |

$i$ | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |

${N}_{i}$ | 70 | 90 | 110 | 75 | 100 | 95 | 85 | 110 | 105 | 95 | 115 |

${o}_{i}$ | 400 | 250 | 250 | 400 | 250 | 250 | 250 | 250 | 250 | 250 | 250 |

${b}_{i}$ | 18 0 | 120 | 120 | 180 | 120 | 120 | 120 | 120 | 120 | 120 | 120 |

$k$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

${s}_{k}\text{}\left[\mathrm{min}\right]$ | 0 | 120 | 0 | 0 | 0 | 150 | 150 | 150 | 150 | 0 |

$k$ | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

${s}_{k}\text{}\left[\mathrm{min}\right]$ | 0 | 0 | 0 | 80 | 0 | 150 | 0 | 0 | 0 | 0 |

$k$ | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |

${s}_{k}\text{}\left[\mathrm{min}\right]$ | 0 | 0 | 120 | 120 | 0 | 0 | 0 | 0 | 150 | 80 |

$k$ | 31 | 32 | 33 | 34 | 35 | 36 | 37 | |||

${s}_{k}\text{}\left[\mathrm{min}\right]$ | 80 | 80 | 80 | 0 | 0 | 0 | 0 |

Crew | Captain | FO | Route (Crew Pairing) Plan | $\mathbf{Time}\text{}\mathbf{on}\text{}\mathbf{Duty}\text{}\left[\mathbf{min}\right]$ | Color |
---|---|---|---|---|---|

6 | 2 | 1 | 15-18-21 | 475 | |

12 | 3 | 10 | 12-17-20 | 490 | |

13 | 4 | 3 | 6-10 | 270 | |

19 | 5 | 11 | 3-4-9-16 | 570 | |

24 | 7 | 9 | 13-19 | 430 | |

31 | 9 | 5 | 2-7-11 | 625 | |

35 | 10 | 7 | 1-5-8-14 | 635 |

$k$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

${s}_{k}\text{}\left[\mathrm{min}\right]$ | 0 | 345 | 0 | 0 | 325 | 150 | 150 | 150 | 150 | 0 |

$k$ | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

${s}_{k}\text{}\left[\mathrm{min}\right]$ | 410 | 0 | 270 | 345 | 410 | 325 | 325 | 325 | 325 | 270 |

$k$ | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |

${s}_{k}\text{}\left[\mathrm{min}\right]$ | 0 | 0 | 120 | 120 | 0 | 270 | 0 | 0 | 345 | 345 |

$k$ | 31 | 32 | 33 | 34 | 35 | 36 | 37 | |||

${s}_{k}\text{}\left[\mathrm{min}\right]$ | 345 | 345 | 345 | 410 | 410 | 410 | 410 |

FRA | MUC | AGP | OPO | VCE | |||||
---|---|---|---|---|---|---|---|---|---|

CAP | FO | CAP | FO | CAP | FO | CAP | FO | CAP | FO |

1,2,3,4 | 1,2,3,10,12 | 6,7,8 | 4,6,8,9 | 10 | 7 | 9 | 5 | 5 | 11 |

Crew | Captain | FO | Route Plan | $\mathbf{Duty}\text{}\mathbf{Time}\text{}\left[\mathbf{min}\right]$ | Color |
---|---|---|---|---|---|

6 | 2 | 1 | 15 | 440 | |

12 | 3 | 10 | 12 | 340 | |

13 | 4 | 3 | 6-10 | 270 | |

19 | 5 | 11 | 3-4-9-16 | 570 | |

21 | 6 | 6 | 17-20 | 325 | |

24 | 7 | 9 | 13-19 | 430 | |

26 | 8 | 3 | 18-21 | 670 | |

31 | 9 | 5 | 2-7-11 | 675 | |

35 | 10 | 7 | 1-5-8-14 | 635 |

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**MDPI and ACS Style**

Graf, V.; Teichmann, D.; Dorda, M.; Kontrikova, L. Dynamic Model of Contingency Flight Crew Planning Extending to Crew Formation. *Mathematics* **2021**, *9*, 2138.
https://doi.org/10.3390/math9172138

**AMA Style**

Graf V, Teichmann D, Dorda M, Kontrikova L. Dynamic Model of Contingency Flight Crew Planning Extending to Crew Formation. *Mathematics*. 2021; 9(17):2138.
https://doi.org/10.3390/math9172138

**Chicago/Turabian Style**

Graf, Vojtech, Dusan Teichmann, Michal Dorda, and Lenka Kontrikova. 2021. "Dynamic Model of Contingency Flight Crew Planning Extending to Crew Formation" *Mathematics* 9, no. 17: 2138.
https://doi.org/10.3390/math9172138