# Determining the Number of Passengers for Each of Three Reverse Pyramid Boarding Groups with COVID-19 Flying Restrictions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Assumptions and Metrics for Evaluation of Boarding Group Assignments of Passengers

#### 3.1. Assumptions for Passengers’ Social Distancing and Luggage

#### 3.2. Metrics Used to Evaluate the Reverse Pyramid Method with Various 3-Passenger Groups

_{p}is the row in which passenger p has a seat, RowTime

_{pr}is the time that passenger p spends in row r (this duration begins when passenger p begins to enter row r and concludes when passenger p begins to leave row r; this convention was chosen because a passenger’s nose and mouth are at the front of the passenger), p’ = passenger boarding before passenger p, AisleSeat

_{p’r}is 1 if passenger p’ has an aisle seat in row r and 0 otherwise, and WindowSeat

_{p’r}is 1 if passenger p’ has a window seat in row r and 0 otherwise.

## 4. Methods

#### 4.1. Agent-Based Model Implementation

#### 4.2. Adaption of Reverse Pyramid for Social Distancing and Unequal Group Sizes

- g1 > 0, g2a > 0, integers.
- g2a ≤ g1 (within a row, window seat passengers board before aisle seat passengers).
- g1 ≤ 29 for a 30-row airplane (because otherwise there are not three boarding groups).

#### 4.3. Full Grid Search for Finding a Good Starting Point for the Local Search

Algorithm 1. Full-grid-search | |

in: | number of simulation trials per data point noSimulations |

out: | boarding metrics for all group configurations m |

1: | forg1 = 1 to 29 do |

2: | for g2a = 1 to g1 do |

3: | m[g1][g2a] ← ABMSimulation(g1, g2a, noSimulations) |

4: | end for |

5: | end for |

6: | |

7: | returnm |

#### 4.4. Local Search Optimization Method

Algorithm 2. Local Search Optimization Method | |

in: | initial boarding group configuration <g1′, g2a’>; weights of average boarding time, average aisle seat risk duration, and average window seat risk duration w1, w2 and w3; number of simulation trials per condition tested noSimulations |

out: | group configuration having the minimum weighted average duration; the group’s weighted average duration |

{initialize the boarding metrics for all group configurations with NIL} | |

1: | forg1 = 1 to 29 do |

2: | for g2a = 1 to g1 do |

3: | m[g1][g2a] ← NIL |

4: | end for |

5: | end for |

6: | |

7: | m[g1′][g2a’] ← ABMSimulation(g1′, g2a’, noSimulations) {boarding metrics determined using NetLogo simulations} |

8: | |

9: | minWeightedAverage ← w1 * duration(m[g1′][g2a’]) + w2 * aisleRisk(m[g1′][g2a’]) + w3 * windowRisk(m[g1′][g2a’]) |

10: | minWeightedAverageConfiguration ← <g1′, g2a’> |

11: | |

12: | Repeat |

13: | neighborMinWeightedAverage ← ∞ |

14: | neighborWithMinWeightedAverage ← NIL |

15: | <x, y> ← minWeightedAverageConfiguration |

16: | |

17: | for i in {-1, 0, 1} do |

18: | for j in {-1, 0, 1} do |

19: | if i != 0 or j != 0 then |

20: | if (x + i ≥ y + j) and (x + i ≥ 1) and (y + j ≥ 1) and (x + i ≤ 29) then |

21: | if m[x + i][y + j] = NIL then {avoid reanalyzing configurations} |

22: | m[x + i][y + j] ← ABMSimulation(x + i, y + j, noSimulations) {store the boarding metrics} |

23: | |

24: | neighborWeightedAverage ← w1 * duration(m[x + i][y + j]) + w2 * aisleRisk(m[x + i][y + j]) + w3 * windowRisk(m[x + i][y + j]) |

25: | |

26: | if neighborWeightedAverage < neighborMinWeightedAverage then |

27: | neighborMinWeightedAverage ← neighborWeightedAverage |

28: | neighborWithMinWeightedAverage ← <x+i, y+j> |

29: | end if |

30: | end if |

31: | end if |

32: | end if |

33: | end for |

34: | end for |

35: | |

36: | if neighborMinWeightedAverage ≤ minWeightedAverage then |

37: | minWeightedAverage ← neighborMinWeightedAverage |

38: | minWeightedAverageConfiguration ← neighborWithMinWeightedAverage {update the best configuration} |

39: | end if |

40: | |

41: | untilneighborMinWeightedAverage > minWeightedAverage |

42: | |

43: | returnminWeightedAverageConfiguration, minWeightedAverage |

## 5. Numerical Simulation Results

#### 5.1. Simulation Results for w_{1} = 100%, w_{2} = 0%, and w_{3} = 0%

#### 5.2. Simulation Results for w_{1} = 0%, w_{2} = 100%, and w_{3} = 0%

_{1}= 0%, w

_{2}= 100%, and w

_{3}= 0%, determined for each full grid combination of g1 and g2a examined. Most of the lowest (dark blue) values for F were encountered when g1 ranged between 15 and 28 and g2a ranged between 10 and 19. The highest values of F are depicted in light yellow and were obtained for any values of g1 that were combined with values of g2a equal to either 1 or 29.

^{2}(i.e., g2a squared). We show this using proof by induction:

- (1)
- Assume: Notation: E(n) = expected number of aisle seat risk encounters by group 2 passengers when n = g2a. E(g2a) = expected number of aisle seat risk encounters by group 2 passengers = (g2a)
^{2}. - (2)
- E(1) = (1)
^{2}= 1 as explained above, as exactly one of the two row 30 aisle seat passengers is seated before the other. - (3)
- E(g2a + 1) = E(g2a) + the expected number of aisle seat risk encounters by group 2 passengers seated in row (30 – g2a). Each of the two group 2 passengers sitting in row (30 − g2a) can be expected to be passed, on average, by g2a passengers from the higher numbered rows and also exactly one of the row (30 − g2a) group 2 passengers will be passed by the other. Consequently, E(g2a+1) = E(g2a) + 2 * g2a + 1 = (g2a)
^{2}+ 2 * g2a + 1 = (g2a + 1)^{2}, thus completing the proof.

^{2}. In total, the expected number of aisle seat risk encounters would be: (g2a)

^{2}+ (30 − g2a)

^{2}. This value would be minimized when g2a is equal to 15.

#### 5.3. Simulation Results for w_{1} = 0%, w_{2} = 0%, and w_{3} = 100%

#### 5.4. Simulation Results for w_{1} = 60%, w_{2} = 35%, and w_{3} = 5%

#### 5.5. Summary and Discussion of Simulation Results

## 6. Concluding Remarks

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 5.**The graphical user interface (GUI) for the agent-based model in NetLogo. (view in the case of reverse pyramid with 3 groups and g1 = 6 and g2a = 3).

**Figure 10.**The performance (in colors) of the g1 and g2a combinations when the objective function F was determined by weights w

_{1}= 100%, w

_{2}= 0%, and w

_{3}= 0%.

**Figure 11.**The starting solution (24, 23), the area explored by the local search method and the best solution (25, 24) for weights w

_{1}= 100%, w

_{2}= 0%, and w

_{3}= 0%.

**Figure 12.**The reverse pyramid scheme for the best local search solution G1 = 50, G2 = 58, and G3 = 12 for weights w

_{1}= 100%, w

_{2}= 0%, and w

_{3}= 0%.

**Figure 13.**The performance of the g1 and g2a combinations for w

_{1}= 0%, w

_{2}= 100%, and w

_{3}= 0%.

**Figure 14.**The starting solution, the area explored by the local search method, and the best solution (15, 15) for weights w

_{1}= 0%, w

_{2}= 100%, and w

_{3}= 0%.

**Figure 15.**The reverse pyramid scheme for the best local search solution G1 = 30, G2 = 60, and G3 = 30 for weights w

_{1}= 0%, w

_{2}= 100%, and w

_{3}= 0%.

**Figure 16.**The performance of the g1 and g2a combinations for w

_{1}= 0%, w

_{2}= 0%, and w

_{3}= 100%.

**Figure 17.**The performance of the g1 and g2a combinations for w

_{1}= 60%, w

_{2}= 35%, and w

_{3}= 5%.

**Figure 18.**The reverse pyramid scheme for the best local search solution G1 = 32, G2 = 60, and G3 = 28 for weights w

_{1}= 60%, w

_{2}= 35%, and w

_{3}= 5%.

Boarding Groups | Case | Weights | Local Search Results | |||||||
---|---|---|---|---|---|---|---|---|---|---|

w1 | w2 | w3 | g1 | g2a | F | Boarding Time | Aisle Seat Risk | Window Seat Risk | ||

Unequal | C1 | 100% | 0% | 0% | 25 | 24 | 891.2 | 891.2 | 2338.0 | 8768.1 |

C2 | 0% | 100% | 0% | 15 | 15 | 1745.8 | 907.7 | 1745.8 | 7192.3 | |

C3 | 0% | 0% | 100% | 15 | 15 | 7192.3 | 907.7 | 1745.8 | 7192.3 | |

C4 | 60% | 35% | 5% | 16 | 16 | 1515.2 | 904.5 | 1749.8 | 7200.5 | |

Equal | C1 | 100% | 0% | 0% | 20 | 10 | 927.6 | 927.6 | 1972.2 | 8298.1 |

C2 | 0% | 100% | 0% | 1972.2 | ||||||

C3 | 0% | 0% | 100% | 8298.1 | ||||||

C4 | 60% | 35% | 5% | 1661.7 |

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## Share and Cite

**MDPI and ACS Style**

Delcea, C.; Milne, R.J.; Cotfas, L.-A.
Determining the Number of Passengers for Each of Three Reverse Pyramid Boarding Groups with COVID-19 Flying Restrictions. *Symmetry* **2020**, *12*, 2038.
https://doi.org/10.3390/sym12122038

**AMA Style**

Delcea C, Milne RJ, Cotfas L-A.
Determining the Number of Passengers for Each of Three Reverse Pyramid Boarding Groups with COVID-19 Flying Restrictions. *Symmetry*. 2020; 12(12):2038.
https://doi.org/10.3390/sym12122038

**Chicago/Turabian Style**

Delcea, Camelia, R. John Milne, and Liviu-Adrian Cotfas.
2020. "Determining the Number of Passengers for Each of Three Reverse Pyramid Boarding Groups with COVID-19 Flying Restrictions" *Symmetry* 12, no. 12: 2038.
https://doi.org/10.3390/sym12122038