# Consideration of Passenger Interactions for the Prediction of Aircraft Boarding Time

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Status Quo

#### 1.2. Scope and Structure of the Document

## 2. Aircraft Boarding Model

#### Complexity Metric

_{seat}: Not used/not booked (C

_{seat}= 0), free (C

_{seat}= 1), and occupied (C

_{seat}= 2). Since the A320 reference is a single-aisle aircraft with three seats located on the left and the right side respectively, these three seats are aggregated to a seat row condition C

_{row, left|right}(C

_{row}). A trinary aggregation of seat conditions per row, according to (1), results in 27 distinct values for C

_{row}. In (1), c represents the number of different seat conditions; in the current case c = 3.

_{row, left|right}= c

^{0}C

_{aisle}+ c

^{1}C

_{middle}+ c

^{2}C

_{window}

_{row}is 26 (= 3

^{0}×

**2**+ 3

^{1}×

**2**+ 3

^{2}×

**2**). Each C

_{row}condition results in a specific number of interactions between passengers and depends on the seat position of the arriving passenger. If, however, the aisle seat is occupied, the next arriving passenger demands a seat shuffle with a minimum of four movements (cf. Reference [48,54]): The passenger in the aisle has to step out, the arriving passenger steps into the row (two steps) and the first passenger steps back into the row. Thus, the interference potential P

_{r, left|right}(P

_{r}) of a specific C

_{row}condition is defined as the expected value of passenger movements (interactions) derived from all probable future seat row conditions (equally distributed).

_{r}(green circles). Here, P

_{r}develops over time and decreases continuously if the seat row is filled up window seat first, followed by middle and aisle seat (P

_{r, step 0}= 8.3, P

_{r, step 1}= 3.5, P

_{r, step 2}= 1). This development results in only three passenger movements, since each passenger can directly step into the row without any interactions. If the boarding sequence starts with the aisle seat and ends with the middle seat, nine passenger movements are necessary, and P

_{r}develops P

_{r, step 0}= 10.5, P

_{r, step 1}= 9.0, and P

_{r, step 2}= 9.

_{r}values increases and results in both additional waiting times and longer boarding time. This effect depends on the number of passengers who have to pass this row or wait in the queue to reach the seats located in front of this row. In a first approximation, only passing passengers with higher seat row numbers will be considered, by counting all seats which are currently free in the concerned row and the rows behind (number is stored as p

_{b,r}and modeled as exponential function with A as scaling factor). In (2), P

_{r}is weighted by indirect interferences and summed up over all seat rows (m) to an aircraft-wide interference potential P. Furthermore, the difference between the minimum and maximum of P is considered as ΔP in the complexity metric as a measurement of convergence (P

_{r}is the expected value in (2)).

_{ri}for passenger i. While iterating over all passengers in the queue, the current passenger i will be virtually seated, which results in a continuous update of future seat row states for subsequently following passengers. Thus, a lower value of function k (3) indicates a lower level of influence between the passengers in the queue.

_{sl}(percentage of seated passengers), interference potential P (blocked seat rows), convergence of interference potential ΔP, and passenger sequence k (consider passengers boarded, but not seated). Figure 2 exhibits the complexity metric using a fast and slow boarding scenario. In the following, the seat load is defined as 1 − f

_{sl}to ensure a consistent behavior of all indicators, which end with a value of zero when the boarding is finished.

## 3. Machine Learning

#### 3.1. Data Transformation

_{sl}. The boarding simulation provides time values for every single indicator of the metric, which could be used as multi-layer input considering current and past values. Since classical statistical methods do not perform well for this given problem, the application of complex and intelligent machine learning approaches is favored. As an example, the boarding dataset could be transformed to a supervised learning problem using P and 1 − f

_{sl}with a window size of one, as a multi-to-one time series (three input features and one output value, see Table 1, cf. Graves et al. [56]).

#### 3.2. Neural Network Models

_{hy}is the hidden-output weight matrix), which basically comprise the parametrization sensibilities of the network. The b terms denote bias vectors as an extraction of the threshold function (e.g., b

_{y}as the output hidden vector) and H is the hidden layer function. Usually H is an element-wise application of the logistic sigmoid function.

_{hy}x(t) + W

_{hh}h(t − 1) + b

_{h})

_{hy}h(t) + b

_{y}

_{xi}x(t) + W

_{hi}h(t − 1) + W

_{ci}c(t − 1) + b

_{i})

_{x}f x(t) + W

_{h}f h(t − 1) + W

_{c}f c(t − 1) + b

_{f})

_{xc}x(t) + W

_{hc}h(t − 1) + b

_{c}

_{xo}x(t) + W

_{ho}h(t − 1) + W

_{co}c(t) + b

_{o})

_{hiddenlayer}), the number of samples propagated through the network for each gradient update (batch-size), the number of trained epochs (n

_{epoch}) and the learning rate (η).

## 4. Simulation Framework and Application

#### 4.1. Simulation Framework

#### 4.2. Scenario Definition

_{sl}, or both are used to predict the output 1 − f

_{sl}. Both scenarios are computed for a given set of start times: 300 s, 400 s, and 600 s after boarding starts. Supervised learning demands for a separation of datasets into training, test and (obligatory) validation data. In our scenario analysis, we have 25,000 separate boarding results for random and individual boarding, and 100,000 boarding results from other strategies. We use 1000 randomly chosen boarding events to train the LSTM and 100 randomly chosen non-compiled /non-trained boarding events to evaluate the prediction.

#### 4.3. Results

_{sl}), a more complex structure is needed, so we set n

_{hiddenlayer}to 120 for these variants (number of hidden LSTM layer next to input and output layer). In the case of uni-variate prediction we set n

_{hiddenlayer}to 40. A higher number of n

_{hiddenlayer}increases the ability of the network to understand complex data structures, but also increases the complexity of learning. AdaGrad works quite well with a more complex network structure, as its learning rate η is 10 times lower than η of Adam, which improves gradient-based learning in multi-branched inlays.

## 5. Summary and Outlook

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Seat row progress (aisle seat right, window seat left) indicating movements/interactions expected (green circle) and realized (blue circle).

**Figure 2.**Complexity metric to evaluate the boarding process and progress: Seat load (

**top left**), interference potential (

**top right**), boarding sequence (

**bottom left**), and convergence of interference potential (

**bottom right**).

**Figure 3.**Long-term interdependencies in RNN models encompassing a connection of input vectors x(0) and x(1) to the hidden output vector h(t + 1).

**Figure 5.**Prediction of seat load progress with LSTM model for two exemplary boarding events (prediction is based on the progress reached after 400 s).

Time | Input A | Input B | Input C | Output |
---|---|---|---|---|

t | X_{A} | X_{B} | X_{C} | Y |

0 | - | - | P(0) | 1 − f_{sl} (1) |

1 | P(0) | 1 − f_{sl} (1) | P(1) | 1 − f_{sl} (2) |

2 | P(1) | 1 − f_{sl} (2) | P(2) | 1 − f_{sl} (3) |

3 | P(2) | 1 − f_{sl} (3) | P(3) | prediction |

Scenario | Boarding Strategy Learned | Boarding Strategy Predicted | Input from Complexity Metric | Prediction Start Time t (s) |
---|---|---|---|---|

A | random | random | {P, 1 − f _{sl},[P, 1 − f _{sl}]} | {300, 400, 500} |

B | all (random, block, back-to-front, outside-in, reverse pyramid, individual) | individual |

Scenario | Input (Uni-Variate) | Input (Uni-Variate) | Input (Multi-Variate) |
---|---|---|---|

A | [1 − f_{sl}] | [P] | [P, 1 − f_{sl}] |

300 s | NaN_{t = 1h 28min} | NaN_{t = 1h 43min} | 289.8 s _{t = 3h 02min} |

400 s | 536.1 s _{t = 0h 30min} | NaN_{t = 1h 01min} | 143.7 s _{t = 2h 38min} |

500 s | 301.8 s _{t = 0h 32min} | NaN_{t = 0h 49min} | 122.3 s _{t = 2h 12min} |

B | [1 − f_{sl}] | [P] | [P, 1 − f_{sl}] |

300 s | 603.1 s _{t = 1h 13min} | NaN_{t = 1h 19min} | 168,1 s _{t = 7h 38min} |

400 s | 292.1 s _{t = 1h 12min} | 389.9 s _{t = 1h 10min} | 77.3 s _{t = 8h 01min} |

500 s | 246.0 s _{t = 1h 08min} | 412.4 s _{t = 1h 11min} | 73.3 s _{t = 7h 54min} |

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**MDPI and ACS Style**

Schultz, M.; Reitmann, S. Consideration of Passenger Interactions for the Prediction of Aircraft Boarding Time. *Aerospace* **2018**, *5*, 101.
https://doi.org/10.3390/aerospace5040101

**AMA Style**

Schultz M, Reitmann S. Consideration of Passenger Interactions for the Prediction of Aircraft Boarding Time. *Aerospace*. 2018; 5(4):101.
https://doi.org/10.3390/aerospace5040101

**Chicago/Turabian Style**

Schultz, Michael, and Stefan Reitmann. 2018. "Consideration of Passenger Interactions for the Prediction of Aircraft Boarding Time" *Aerospace* 5, no. 4: 101.
https://doi.org/10.3390/aerospace5040101