Airline Overbooking Reparation Modeling Using Binomial Distribution

Airline Overbooking Reparation Modeling Using Binomial Distribution Zachariah Alzubi and Zhaojie Pan  Abstract The airline industry is a large and constantly changing financial sector with a value of 359.3 billion US dollars. Airlines need to plan and manage their resources effectively to maintain profitability, and one of the most important and common ways to do so is by overbooking tickets. This involves determining the probability of a passenger missing a flight and selling more tickets than the number of seats available. The study described here aims to determine the maximum compensation that should be offered to passengers who are unable to board a flight due to overbooking. The scope of the study includes modeling the number of empty seats on a flight using a binomial distribution and calculating the expected value of the number of empty seats. The maximum compensation is calculated by dividing the expected loss per passenger by the number of extra tickets sold. The results of this study are expected to provide valuable insights for airlines and inform their decisions regarding overbooking compensation by determining the maximum compensation that could theoretically be offered while still remaining fiscally feasible. The experiment was conducted using the Python programming language and Jupyter Notebook environment, with the math library used for the binomial distribution calculations and custom functions for the expected value and maximum compensation calculations. Table of contents and Nomenclature I. Introduction - A.………………………………………………………….. Background B.………………………………………………………….. Objectives C.………………………………………………………….. Scope II. Analysis - A.………………………………………………………….. Binomial Distribution B…………………………………………………………... Expected Value Calculation C.………………………………………………………….. Maximum Compensation Calculation III. Experimental Equipment and Procedure - A…………………………………………………………... Equipment B.………………………………………………………….. Algorithms and Software C…………………………………………………………... Procedures D…………………………………………………………... Reproducibility IV. Results and Discussion - A…………………………………………………………... Results B.………………………………………………………….. Interpretation of Results C.………………………………………………………….. Comparison to Similar Studies V. Conclusion - A…………………………………………………………... Summary of Results VI. References - A…………………………………………………………... List of cited sources VI. Appendices - Nomenclature Variables N - Total number of seats on the flight p - Probability of a passenger showing up L - Dollar loss per empty seat m - Number of extra tickets sold y - Number of empty seats R - Maximum reparation to be offered EV - Expected value per seat C(n, y) - Binomial coefficient (n choose y) P(y) - Binomial probability mass function (PMF) Mathematical Symbols ^ - Exponentiation ! - Factorial ∑ - Summation C(n, y) = n! / (y! * (n - y)!) - Binomial coefficient formula P(y) = C(n, y) * (p^y) * ((1 - p)^(n - y)) - Binomial PMF formula EV = ∑y * P(y) - Expected value calculation formula R = (EV * L) / m - Maximum compensation calculation formula Introduction Background The airline industry is a rapidly growing and dynamic financial sector worth nearly 359.3 billion U.S. dollars. It is an industry that requires careful planning and efficient management of resources to maintain profitability with razor-thin margins. One critical aspect of this is taking advantage of the fact that for any flight, there is a mathematically attainable figure that represents the likelihood of a passenger not showing up for their flight. With this percentage in mind, airlines can find an optimal amount of tickets to oversell and still be able to more than likely offer every passenger that shows up to their flight a seat. Although airlines can oversell tickets in the most statistically backed fashion, there is still always the possibility of an extra person showing up and having to be denied a seat. In this case, airlines must be able to offer cash reparations to compensate the upset customer. This leads us to the objective of our study. Objectives The objective of this study is to determine the maximum compensation that should be offered to passengers who are unable to board a flight due to overbooking. This compensation is an important consideration for airlines as it directly affects the net-earnings / net-profit. Scope We used the binomial distribution to model the number of empty seats on a flight, given a certain probability (p) of each passenger showing up. To perform this analysis, we consider a range of probabilities (from 90% to 100%) and oversold seats (from 1 to 10). These ranges can be easily tweaked for any desired combination. For each combination we use the binomial distribution to calculate the expected number of empty seats and determine the maximum compensation that should be offered to each passenger. This compensation is calculated based on the expected value of the number of empty seats, the cost per empty seat (L), and the number of extra tickets sold (m). The results of this analysis are then used to calculate the average compensation that should be offered to each passenger for all combinations of probabilities and oversold seats. The results of this study will provide valuable insights for airlines and inform decisions on the appropriate compensation to offer in cases of overbooking. It is worth noting that we also constructed an alternative algorithm that only allows for only set values of N, p , L, and m. This algorithm returns results with negligible margins from the first algorithm ( usually within five dollars discrepancy) when run with comparable L and m values. This confirmed the validity of our findings when run with ranging values for N and p. Analysis Binomial Distribution A binomial distribution is a statistical distribution that describes the number of successes in a fixed number of independent trials, each with the same probability of success. In the context of this experiment, a success is defined as a passenger showing up for their flight. The probability mass function (PMF) for a binomial distribution is given by the formula: P(y) = C(N, y) * p^y * (1 - p)^(N - y) where: • N is the total number of trials, • y is the number of successful trials (number of passengers who show up), • p is the probability of success for each trial, • C(N, y) is the number of combinations of N things taken y at a time, given by the formula: C(N, y) = N! / (y! * (N - y)!) Expected Value Calculation The expected value of a random variable is the average of its possible values, weighted by their probabilities. In this experiment, the expected value of the number of empty seats on a flight can be calculated by summing up the product of each possible number of empty seats and its corresponding probability: EV = ∑(y * P(y)) where y ranges from 0 to N, and P(y) is the probability of y empty seats as calculated from the binomial distribution PMF. Maximum Compensation Calculation The maximum compensation offered to each passenger is calculated as the expected loss per passenger, divided by the number of extra tickets sold: R = (EV * L) / m where:  • EV is the expected value of the number of empty seats, • L is the loss incurred by each empty seat, • m is the number of extra tickets sold. This formula represents the maximum compensation that should be offered to each passenger to cover the expected loss due to overbooking. In this study, we use a double for loop to iterate over a range of probabilities(from 90% to 100%) and oversold seats(from one to ten ), and for each combination, we use the binomial distribution and expected value calculation to determine the maximum compensation. The results of these calculations are then used to calculate the average compensation that should be offered to each passenger. It is once again worth noting that we also constructed an alternative algorithm that only allows for only set values of N, p , L, and m. This algorithm returns results with negligible margins from the first algorithm ( usually within five dollars discrepancy) when run with comparable L and m values. This confirmed the validity of our findings when run with ranging values for N and p. Experimental Equipment and Procedure Equipment The code was written in the Python programming language and executed using the Jupyter Notebook environment. Algorithms and Software To perform the calculations involved in the experiment, the math library was used for the binomial distribution calculations. The comb function was utilized to calculate the number of combinations of a set of elements. The binomial distribution and expected value calculations were performed using a custom function. The maximum compensation calculation was performed using another custom function that utilized the binomial distribution and expected value calculations. Procedures The experiment involved looping over a range of ‘show-up’ probabilities (90% to 100%) and oversold seats (1 to 10 seats) and calculating the maximum compensation offered to each passenger. The binomial distribution and expected value calculations were performed using the custom functions mentioned in the Algorithms and Software section. The results were then used to calculate the average maximum compensation offered to passengers over all combinations. The experimental procedure consisted of running two different blocks of code side-by-side in Jupyter Notebook. The first block of code calculates the binomial probability, expected value, and maximum compensation offered to each passenger for set elements N, p, L , and m. The second block of code runs a loop over a range of probabilities (90% to 100%) and oversold seats (1 to 10 seats), while maintaining the same L and m, then calculating the average compensation offered in each scenario. In both blocks of code, the function binomial_prob calculates the binomial probability, the function expected_value calculates the expected value of the number of empty seats, and the function max_compensation calculates the maximum compensation offered to each passenger. These functions are used to provide the results and data for analysis and both return very similar results. Reproducibility The experiment is open to peer review and can be easily reproduced by someone familiar with the Python programming language and the concepts of binomial distribution and expected value calculation. Results and Discussion Results The first block of code with N = 100, p = 0.95, L = 100, and m = 5 returns the maximum compensation to offer is 1900 dollars. In the pursuit of full thoroughness, while the code runs through the expected value function, it prints out the binomial probability: ‘bp’ , of the number of riders associated : ‘i’. When we plot this relationship between the probability of ‘i’ people showing up versus ‘i’ people with this code : We get this graph: We assumed the graph would peak at 95 percent ( i.e the most likely outcome for total riders would be 95 since p= 0.95 and N= 100), which is very clearly demonstrated in the graph confirming the validity of our algorithm i.e The likelihood of 95 people exactly showing up is approximately 18 percent, which is the highest calculated percentage and in turn the most likely event. In the second block of code, the code ranges from 90 - 100 percent probability and oversold seats range from 1-10. The code returns the average maximum compensation over all of the combinations to offer. This ends up being 1911.5 dollars which is a mere 0.6% dollar amount greater than the previous simulation, i.e they agree within a very small margin of error. This discrepancy can however be explained by the fact that the second algorithm optimizes for the ‘worst-case’ scenario p value between 90 and 100, whereas the first algorithm just assumes it to be 95 percent (which must be very close considering the closeness of the two final reparation values) . Upon computing the real worst-case p value to be .95575 , and plugging that into the original algorithm, we found the two algorithms to exactly agree on 1911.5 dollars.  Interpretation of Results The results provide an insight into how the binomial function can be used to determine the optimal revenue for an airline after calculating the probability of the number of passengers that will show up when overbooking a specific route. The probability is modeled with a binomial function, assuming that each show or no-show sample is independent of each other. The experiment also highlights the importance of considering the value of the ticket, the voucher cost, the number of seats, the probability rate of a passenger showing up, and the number of additional seats when using the binomial function to calculate the overbooking in an airline. Some potential weaknesses of the constructed model are as follows: • Dependence on the accuracy of input data - If the input data used in the model is incorrect or not suited to the specific flight, it can lead to unreliable results. • Limited scope of analysis - The model may only consider a narrow range of factors affecting overbooking, ignoring important variables such as weather conditions or flight delays. • Assumption of independence - The model assumes that each show or no-show decision made by a passenger is independent of others, which may not always be the case in real-world scenarios. A good example would be families traveling together or just the standard group size of travelers • Simplistic approach - The model may only consider a limited number of variables, such as ticket price and voucher cost, ignoring other important factors such as passenger behavior and airline policies. • Limited ability to handle non-linear relationships - The model may not be able to effectively analyze complex relationships between variables, such as the relationship between ticket price and passenger behavior. • Limited ability to incorporate historical data - The model may not be able to effectively analyze patterns and trends in overbooking patterns over time. Comparison to Previous Studies The study compares well with a similar investigation done by Cory Simon, where he uses a normal approximation of a binomial distribution to allocate fixed parameters such as the number of seats per aircraft, the probability of a passenger and expands to a range of additional tickets beyond capacity sold the price per ticket, and the price of the voucher. However, the present study expands on Simon's work by creating a function that allows the passing of the value of the ticket, voucher cost, number of seats, probability rate of passengers showing up, and number of additional seats. This allows for a more comprehensive analysis of the problem. The study also highlights the limitations of the binomial function model, as there are four moving variables (ticket sale price, voucher cost, probability of passengers showing up, and number of seats per aircraft) that could impact the results. This underlines the need for further investigations to provide a more comprehensive understanding of the model and its application in aviation operations. Conclusion Summary of Results In conclusion, our study found that the binomial distribution function can be used to determine the optimum revenue for an airline by calculating the probability of the number of passengers that will show up when overbooking a specific route. Our results showed that the ideal voucher cost should be less than a certain amount, and that overbooking a certain number of tickets would result in the maximum total profit. The results of the analysis provide a basis for understanding the model when applying analytics in aviation operations, though there are four variables that can impact the final outcome, including the ticket sale price, voucher cost, probability of passengers showing up, and number of seats per aircraft. With the inputted p, L , m , N of: N = 100 p = 0.95575 L = 100 m=5 We found the optimal maximum reparation voucher value to be 1911.5 dollars. With the inputted value of: L=100 m=5 And ranging values of p and N, We found the optimal maximum reparation voucher value to be 1911.5 dollars. References List of Cited Sources Python - Binomial Distribution - GeeksforGeeks. (2020, June 26). GeeksforGeeks. Retrieved February 5, 2023, from https://www.geeksforgeeks.org/python-binomial-distribution/ What is the binom.pmf() method in Python? (n.d.). Educative: Interactive Courses for Software Developers. Retrieved February 5, 2023, from https://www.educative.io/answers/ what-is-the-binompmf-method-in-python Binomial Distribution: Formula, What it is, How to use it. (n.d.). Statistics How To. Retrieved February 5, 2023, from https://www.statisticshowto.com/probability-and-statistics/ binomial-theorem/binomial-distribution-formula/ Calculating The Cost Of Overbooking Airline Tickets Using a Binomial Function. (n.d.). Calculating the Cost of Overbooking Airline Tickets Using a Binomial Function. Retrieved February 5, 2023, from https://www.linkedin.com/pulse/calculating-cost-overbooking-airline- tickets-using-carlos Overbooking: How to avoid plane rage. (n.d.). Plus Maths. Retrieved February 5, 2023, from https://plus.maths.org/content/overbooking Global airline industry market size 2018-2021 | Statista. (n.d.). Statista. Retrieved February 5, 2023, from https://www.statista.com/statistics/1110342/market-size-airline-industry- worldwide/