Journal Pre-proof Determine the cost of denying boarding to passengers: An optimization-based approach Lijian Chen PII:
S0925-5273(19)30245-2
DOI:
https://doi.org/10.1016/j.ijpe.2019.07.008
Reference:
PROECO 7435
To appear in:
International Journal of Production Economics
Received Date: 5 March 2019 Revised Date:
11 June 2019
Accepted Date: 9 July 2019
Please cite this article as: Chen, L., Determine the cost of denying boarding to passengers: An optimization-based approach, International Journal of Production Economics (2019), doi: https:// doi.org/10.1016/j.ijpe.2019.07.008. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Determine the Cost of Denying Boarding To Passengers: An Optimization-based Approach Lijian Chen Department of MIS, Operations Management, and Decision Sciences, University of Dayton
[email protected] July 31, 2019
Abstract We present an optimization-based method to evaluate the financial impact of denying boarding to the lowest-fare customers under airline network overbooking. Airlines are demanding quantitative support to evaluate the impact on revenue when facing the decision to deny boarding to customers. Although overbooking is a well-regulated operation for decades, it has drawn considerable criticism due to, at least partially, disagreement regarding compensation. Usually, airlines deny boarding to the lowest-fare customers to assure seat availability for customers in higherpaying booking classes. We model such an assurance by chance-constrained optimization, and we gain managerial implications for overbooking practices, for low-cost airlines in particular. We conclude that low-cost airlines are vulnerable regarding revenue growth under network overbooking in comparison to other airlines. We explore possible revenue-improving suggestions, which will ease the effects of network booking for airlines both analytically and numerically. Keywords: Seat Inventory Control, Chance-constrained Optimization, Overbooking, Origin-destination, Fareclass
1 Introduction Overbooking is a common practice of airlines to sell more seats than their actual supply. Such a practice is in response to the fact that a fraction of customers will cancel their reservations at a later time, or will simply not show up at the boarding gates. This practice of overbooking ensures that a minimum number of seats will be empty, i.e., spoilage, resulting in the maximum revenue at a fixed cost. On some occasions, the overbooking may lead to some unpleasant moments for some customers who will be denied boarding. The airline company may ask volunteers to give up 1
their seats in exchange for compensation, e.g., hotel accommodations, and a free flight at a later time. If no volunteers give up their seats, the aircrew will select customers to receive matching compensation as volunteers. These volunteers or selected customers, who are denied boarding, are called denied-boarding customers. In this research to study the financial impact of overbooking, we do not differentiate between involuntary and voluntary denied-boarding customers because any denied-boarding customer will be subject to compensation. Overbooking has mixed impacts on airline operations. The benefits of overbooking are significant. In Smith et al. [1992], the author reports that for a sold-out flight, the airline may experience a 15% of no-shows if the overbooking are not in place. Davis [1994] reports that overbooking saved American Airlines an estimated $1.4 billion over a three-year period. Thus, airlines including United, Delta, and American, keep their overbooking because overbooking will maximize revenue by minimizing spoilage, i.e., seats fly empty. Meanwhile, overbooking may lead to unpleasant events, which usually result in negative brand perceptions and profit loss. In the US, there are more than 50,000 denied-boarding customers annually; as a result, overbooking is well regulated by federal guidelines. For example, as of the year 2017, if the domestic traveler arrives between one and two hours later than planned, the airline must pay the passenger twice the amount of the one-way fare to his destination, up to $675. If the passenger is delayed for more than two hours, the airline must pay him four times the one-way fare, up to $1,350. In addition to the U.S. market, the compensations to the denied-boarding customers are all progressive worldwide. For example, airlines are required to pay denied-boarding customers 250 EUR if the remaining traveling distance is 1500km or less; 400 EUR if the remaining distance is more than 1500km within the European Union and less than 3500km; for all the flights which need to travel more than 3500km, the denied-boarding customer is entitled to a 600 EUR compensation. For Asian countries excluding China, the compensation for denied-boarding customers are progressive as well. For JAL and ANA, each denied-boarding customer will be paid 10000 Yen to change a flight departing the same day and 20000 Yen to those who agree to depart the following day or later. Essentially, all denied-boarding customers worldwide, will be compensated in a similar way. Our research, which is primarily studied on the U.S. market practice, could be applied to other markets as well. There have been incidents caused by overbooking over the years. During a five-year span ending in 2016, United Airlines alone denied boarding to over 42,500 customers involuntarily from overbooked flights. Some incidents, which were perceived as overbooking-related, have been public relations (PR) disasters. Concerned about these mixed impacts of overbooking, Southwest and Jet Blue have publically announced that they would terminate their overbooking entirely, while United, Delta, and American continue to overbook their flights. In the aftermath of the UA3411 2
incident, United Airlines has pledged to offer up to $10,000 compensation to denied-boarding customers, i.e., a 640% increase from the federal guidelines. Such activities of offering a compensation spike or terminating overbooking entirely indicate that airlines are overreacting to these incidents. Major airlines are divided on the issue of overbooking: it is difficult to determine the financial impact of denying boarding to customers due to many complex behavioral and psychological factors. Most of these factors, such as customer dissatisfaction and poor perceptions, are not quantitatively measurable. That is why the revenue management research on overbooking mostly concerns efforts to control the risk of denying boarding to customers by avoiding excessive overbooking. The most recent research regarding overbooking is in Ely et al. [2017] and references therein. We realize that most research papers involve the admission policies regarding booking requests (e.g., Erdelyi and Topaloglu [2009, 2010]). In Kunnumkal and Topaloglu [2011], the authors try to calculate the bid price using the stochastic gradient of the expected revenue. In Huang et al. [2013], authors extend the overbooking from a single-leg problem to a problem of parallel flights. These efforts are intended to address the seat allocation problem with overbooking using a better admission policy, while our research is intended to evaluate the reactive impact of denying boarding to customers. The federal regulations can barely help in this aspect because following the regulatory guidelines would only provide legal grounds for airlines rather than easing customer dissatisfaction and poor perceptions. The operational cost analysis of denying boarding to customers is not adequately addressed, except for some efforts in Suzuki [2006]. The author uses simulations to evaluate the benefit of overbooking for single-leg itineraries. Our research serves the similar purpose that without simulating individual customers’ traveling, we use the difference between the objective functions to calculate the compensations for overbooked flights. This research is to answer the question: “how much will it cost the airline to deny boarding to a customer whenever necessary?” We answer this questions by comparing a pair of wellknown Origin-Destination-Fareclass (ODF) seat inventory control models under demand uncertainty. With overbooking, airlines will allocate more seats than they have and the seat allocation plan will be carried out by booking agencies. We refer the corresponding seat allocation plan as the model of implementation (MoI). In other words, the ODF based overbooking plan has been widely used by airlines in their operations. According to Chen and Homem-de Mello [2010], the objective function of the ODF model will be the expected revenue without taking no-show and denying boarding to customers into account. Airlines will also apply the nesting method, which assigns a lower-class seat to a request from a higher booking class when its own seat inventory is empty. For example, a business traveler will be nearly guaranteed to book a coach seat at the business fare when the business seats are sold out. With the nesting operation, the seats that are allocated to different higher-paying classes will form a pool of seats rather than segmented allocations. 3
The another ODF model is to study overbooking based on the assurance of seat availability for higher-paying customers. In this model, we adopt chance-constrained optimization. Its objective function is also the expected total revenue and the solution is a seat allocation plan. The primary difference of this seat allocation plan from the model of implementation is that, with a high probability, there will be no denied-boarding customers under this seat allocation because we intentionally allocate more seats to the higher-paying customers. The optimization with the chance-constraint would be efficiently solved with the global solution, using the method in Chen [2017]. The solution of the chance constrained ODF model will only serve the purpose of comparison rather than real-world implementation. Thus, we refer the chance constrained ODF model as the model of comparison (MoC). We note that the MoC model is not implementable because it intentionally allocates more seats to higher-paid customers. This is clearly not optimal in reality. The purpose of MoC, combined with MoI, is to determine the fair amount of compensation to denied-boarding customers when overbooking occurs. It is a straightforward result that the optimal value of MoC is no more than its counterpart of MoI because MoC models the activity of denying boarding to lowest-fare customers by imposing the chance constraints and intentionally allocating more seats for higher-paying customers. That is, the optimal value of MoI is the total expected revenue without penalty while the optimal value of MoI is the total expected revenue with penalty. The difference between the two optimal values will be the impact of denying boarding to a necessary number of lowest-fare customers from overbooked flights. The primary contribution of this paper is to model denying boarding to lowest-fare customers by chance-constrained optimization and we thereby draw a conclusion on the reason that low-cost airlines are reluctant to embrace overbooking followed by revenue-improving suggestions. The chance-constrainted optimization will serve as a worst-case scenario analysis because we eliminate denying boarding with a high probability level by intentially taking seats away from the lowest-fare customers. We provide a quantitative analysis for the impact of denying boarding to customers that the difference in the expected revenues of the two models will be the cost of compensation for the denied-boarding customers. The obtained value of compensation, normalized by the number of denied-boarding customers, will provide a critical decision support for airlines because it involves the cost of overbooking per denied-boarding customer. With the cost of overbooking per denied-boarding customer and the Federal regulatory requirements regarding compensating denied-boarding customers, the airline would make a rational decision rather than to offer either compensation or terminate overbooking entirely. In the numerical study, low-cost airlines would be more interested in shutting the overbooking operation down than other major airlines. From an academic point-of-view, we build an optimization model with a closed-form expres4
sion rather than solely relying on case-specific numerical outcomes. Furthermore, in our MoC, the demand of multiple booking classes would be jointly distributed rather than assumed to be independent. Thus, our proposed model is close to reality with less restrictive assumptions. Through an analysis, we would gain insights into the quantitative impact of overbooking in a network setting. Using our method, airlines would be able to selectively overbook flights in order to minimize both spoilage and negative customer perceptions at the same time. We organize the remainder of the paper as follows. In Section 2, we introduce our notations and model formulations with both demand and no-show uncertainties. The model is intended to allocate seats in an origindestination setting. In Section 3, we compare chance-constrained optimization against network stochastic seat allocations with overbooking. We use the difference between the expected revenues from both models as the cost of overbooking caused by compensating denied-boarding customers. Based on this value, the airline would determine how to selectively overbook flights in the network in order to minimize spoilage and ease negative customer perceptions. In Section 4, we present the numerical results to support our analysis in the previous sections to draw conclusions in Section 5.
2 Notations and models We consider a network of flights involving p booking classes of customers. On a particular day, each customer requests one out of n possible itineraries; thus, we have r := np itinerary-fare class combinations. The booking process is realized over a time horizon of length τ . Let {Njk (t)} denote the point process generated by the arrivals of class-k customers who require itinerary j. In most revenue management literature, {Njk (t)} would be a Poisson process, which can be approximated by a normal random variable whenever the arrival rate is greater than 5, or a fitted distribution from the empirical distribution. Throughout this paper, however, we do not make these assumptions, except when explicitly stated. Also, we need to assume that there is at most one unit of demand per time period. The demand for the itinerary-class (j, k) over the whole horizon is denoted by ξjk , and we denote ξ as the whole vector (ξjk ). ξ, in most cases, is a random variable with finite support. The network is composed of m leg-cabin combinations, with capacities c := (c1 ; . . . ; cm ), and is represented by an (m × np)-matrix A := (ai,jk ). The notation m also represents the number of flights operating in the network. The entry ai,jk ∈ {0, 1} indicates whether class k customers use leg i in itinerary j. We denote xjk as the decision variable corresponding to the number of seats to be allocated to class k in itinerary j. Whenever an itinerary-class pair (j, k) is accepted, the revenue corresponding to the fare fjk occurs. A customer’s request is rejected if no seats are available for 5
this itinerary-class, in which case no revenue is realized. The vectors of the decision variables and fares are denoted respectively by x = (xjk ) and f := (fjk ). The seat allocation model requires solving an optimization problem in order to find the initial allocations before the booking process starts. The classical seat allocation model for the network problem is written as follows: max{f T x : Ax ≤ c, x ≤ E[ξ], x ≥ 0} In this model, the random demand ξ is replaced by its expected value to obtain a deterministic model for simple solution. The optimal solution is x∗ = (x∗jk ). Implementation of the resulting policy is rather straightforward. The airline will accept at most x∗jk class k customers in itinerary j. This model is well known in the revenue management literature, such as Williamson [1992] and references therein. The objective function of this model is the upper bound of the expected airline revenue when ξ is a bounded random vector. We note that the demand uncertainty would be incorporated into the seat allocation. We have the following stochastic programming model: max{f T E[min{x, ξ}] : Ax ≤ c, x ≥ 0} where the min operation is component-wise. The objective function of this model is the optimal expected revenue generated from the seat allocation (see Chen and Homem-de Mello [2010]). The objective of this model is to obtain a seat allocation plan to maxmize the expected revenue under the random demands. In comparison to the determinstic model, this model is complex regarding the solution technique and its generated revenue will no less than the counterpart of the deterministic model. The decision x represents the seat allocation plan rather than materialized booking reservations. The constraint Ax ≤ c indicates that the seat allocation plan is subject to the network seat capacity constraints. Modeling the overbooking is rather straightforward. There are two uncertainties: the random demands and the number of no-shows across the network. We use ζ, a random vector of the same dimension of ξ, to represent the number of no-shows for all booking classes. It is reasonable to assume that almost surely, we have: ξ ≥ ζ a.s. where “a.s.” stands for the term “almost surely”. It is true that the no-show ζ depends on the number of seats allocated, i.e., x. However, due to the curse of dimensionality, practitioners and researchers assume a no-show probability without class dependency for the sake of simplifying the calculation regarding the no-show rate. In this paper, we can further assume that the no-show will always be less than any underlying seat allocation from various ODF models because there 6
are only two booking classes: the aggregated higher-paying class, and the lowest-fare class. In Subramanian et al. [1999] and many other papers therein, researchers conclude that the no-show rates would be around to 9%. Thus, we have x ≥ ζ a.s. where x is the seat allocation plan of the ODF model. We remark that the no-show customers are different from the denied-boarding customers. The no-show customers take the full responsibility and give seats voluntarily while the deniedboarding customers are deboarded either voluntarily or involuntarily based on necessity. The noshow rate varies among booking classes. Higher-paying customers consist of individuals or firms with sufficient financial means to afford the ticket prices, and the no-show probability for higherpaying customers exhibits higher values while the lowest-fare customers may be otherwise. Thus, it is likely that there will be more higher-paying customer reservations recycled than the lowestfare customers. However, none of denied-boarding customers is a no-show and airlines will only deny boarding to the lowest-fare customers for an overbooked flight. To overbook flights, airlines allocate more seats than they have. For example, when the airline decides to overbook by 15% on the ith flight of 137 seats available (Boeing 737), the airline will use 158 instead of 137 as the value of ci . To model such an adjustment, we use λ > 1 as such a predetermined value for overbooking, e.g., λ = 1.15 suggests that the airline overbooks by 15%. The value of λ is not subject to constant changes for most airlines because it represents the number of seats which do not exist. A change to the value of λ is a change to an important content of booking policy and such a change would be treated with caution for all airlines. Thus, λ is rather a value than a variable and when overbooking, airlines set the following constraints: Ax ≤ λc. We remark that the problem to determine the overbooking limit λc has been an important overbooking-related question to the revenue management research community for years. In Aydın et al. [2012], you can find a good summary of past research regarding a handful of solutions regarding this problem. In particular, we find the results in Topaloglu et al. [2012], Kunnumkal and Topaloglu [2008], and Frenk et al. [2016] to find an overbooking limit by stochastic dynamic-programmingbased methods. In practice, airline analytical department will use available information including the historical customer demand pattern, the upcoming convention, the sport events, and future weather conditions to forecast the gross customer demand and the number of no-shows. Based on mostly statistical analysis, a time-sensitive overbooking limit will be calculated and implemented at the leg-cabin level. 7
In this research, we impose the constraints (2.1) in the ODF model and assume the possession of the overbooking limits because rather than studying an overbooking policy, we focus on the financial impact of denying boarding to customers, which is the consequence of overbooking. The decision of the overbooking limit has been made. Furthermore, for a set of network capacities of (c1 ; . . . ; cm ), the overbooking limits would be leg-cabin dependent, i.e., (λ1 c1 ; . . . ; λm cm ), λ := (λ1 ; . . . ; λm ) to be consistent with the practice of airlines. For the sake of notation simplification, we continue to use the notation λc in the remainder of this paper without altering the outcome of this research. We have the deterministic seat allocation model with overbooking: max{f T min{x, E(ξ − ζ)} : Ax ≤ λc, x ≥ 0}.
(2.1)
The overbooking is implemented by increasing the capacity c to λc where λ > 1. The difference between this model and the model in Chen and Homem-de Mello [2010] is the available capacities, c and λc. Both models are identical regarding the solution technique and most constraints.
In
Section 4, we present the solution technique of model (2.1), which could be transformed into a linear program. Likewise, the stochastic seat allocation model with overbooking would be: ν ∗ (λ) := max{f T E[min{x, (ξ − ζ)}] : Ax ≤ λc, x ≥ 0}.
(MoI)
where we use ν ∗ (λ) and x∗ (λ) to denote the optimal expected revenue and the optimal seat allocation under a certain overbooking level λ, respectively. Model (MoI) would be solved by the SAA method (see Kleywegt et al. [2002]) where the term “MoI” stands for the expression “Model of Implementation”. The optimal solution of (MoI) is the seat allocation plan, which declines the lowest-fare customers if necessary and its optimal value is the expected total revenue without compensation for denied-boarding customers considered. In this research, we are interested in the quantitative impact of denying boarding to customers for overbooked flights. Since overbooking is intended to sell an excessive number of seats that do not exist, to counter the no-shows, the airlines would start denying boarding to customers in the lowest-fare booking classes to guarantee seats for the higher-paying customers. Let H represent the set of higher-paying booking classes, while E represent the lowest-fare booking classes in the network. It is clear that there will be n lowest-fare classes for n itineraries, and the remaining n(p − 1) booking classes involve higher-paying customers. We use ξH to represent the customer demands of the higher-paying booking classes and ξE to denote the demands of economy booking classes such that ξ := [ξH ; ξE ]. Similarly, AH and AE are m-row matrices that represent the leg-cabin combinations for the higher-paying classes and the lowest-fare class, respectively. The total number of columns of AH and AE is np such that A = [AH AE ]. The no-shows of the booking classes are 8
ζ := [ζH ; ζE ], where ζH and ζE are random vectors of the no-shows for the higher-paying booking classes and the lowest-fare booking class, respectively. The stochastic demands for m flights are: [AH AE ](ξ − ζ) = AH ξH + AE ξE − AH ζH − AE ζE If the seat allocation decision is x, the seats allocated to the high-paying booking classes on m flights would be: [AH 0]x To assure seat availability for the higher-paying customers with a high probability, e.g., 1 − α, we have: P(AH (ξH − ζH ) ≤ [AH 0]x) ≥ 1 − α
(2.2)
where α < 0.5 represents the probability that the higher-paying customers would be denied boarding, and 0 is a matrix of the same dimensions as AE , with all of the components valued at zero. The seat allocation plan x, which satisfies (2.2), will assure seat availability for the higher-paying customers with a probability of 1 − α. We use the following model to evaluate the impact of denying boarding to the lowest-fare customers: νb∗ (λ) := max f T E[min{x, ξ − ζ}] subject to Ax ≤ λc P(AH (ξH − ζH ) ≤ [AH 0]x) ≥ 1 − α
(MoC)
x≥0 whose objective function is the expected revenue when assigning fewer seats to the lowest-fare customers. In this model, the solution is another seat allocation plan. The optimal solutions of (MoC) is the seat allocation plan which intentionally assigns more seats to higher-paying classes in order to assure seat availability for higher-paying customers, with a high probability 1−α where the term “MoC” stands for the expression “Model of Comparison”. The deboarding activity is highly unlikely because the lowest-fare customers will not be admitted in the first place according to the seat allocation plan of (MoC). We use νb∗ (λ) and x∗b (λ) to denote the optimal expected revenue and the optimal seat allocation of model (MoC). We note that x∗b (λ) will not be implemented in reality because it is overly conservative. We now show that model (MoC) would be equivalent to a convex optimization under a mild assumption: Definition 1. A function g(z) ≥ 0, z ∈ Rn is said to be logarithmically concave (logconcave in a short form), if for any z1 , z2 and λ ∈ (0, 1), we have g(λz1 + (1 − λ)z2 ) ≥ [g(z1 )]λ [g(z2 )]1−λ 9
Definition 2. A probability measure defined on the Borel sets of Rn is said to be logarithmically concave (logconcave) if for any convex subsets of Rn : X, Y and λ ∈ (0, 1), we have: P(λX + (1 − λ)Y ) ≥ [P(X)]λ [P(Y )]1−λ where λX + (1 − λ)Y = {z = λx + (1 − λ)y|x ∈ X, y ∈ Y }. Based on these definitions, we have Theorem 1. If ξ is an n-dimensional random variable, the probability distribution of which is logconcave, then the cumulative probability distribution function G(z) := P(ξ ≤ z) is a logconcave function in Rn . The proof of Theorem 1 is in Shapiro et al. [2014], and we omit it. We apply Theorem 1 to show that the chance constraint (2.2) would be equivalently transformed into a convex function. Theorem 2. If the distributions of ξH and ζH are logconcave, model (MoC) would be equivalent to a convex optimization with respect to x. Proof. The objective function of (MoC) is concave function of x to maximize the expected revenue. The constraints Ax ≤ λc are linear constraints. Thus, the only problematic constraint is the chance constraint: P(AH (ξH − ζH ) ≤ [AH 0]x) ≥ 1 − α This constraint is neither convex nor concave. We can develop an equivalent constraint which is convex to replace it.
Since both ξH and ζH are logconcave, ξH − ζH would be logconcave (see
Saumard et al. [2014] for the operations which preserve log-concavity). The matrix AH is a deterministic m-row matrix, and AH (ξH −ζH ) would be a logconcave random vector with the cumulative probability function FH . Therefore, the chance constraint is equivalent to: FH ([AH 0]x) ≥ 1 − α. We take natural log operation from both sides, we have log(1 − α) − log FH ([AH 0]x) ≤ 0 which is a convex function with respect to x. Thus, model (MoC) is equivalent to the following model: max f T E[min{x, ξ − ζ}] subject to: Ax ≤ λc log(1 − α) − log FH ([AH 0]x) ≤ 0 x ≥ 0. We are done. 10
(2.3)
We note that the assumption of both demand and no-show distributions being logconcave is mild. In the literature, the customer demands and no-shows are overwhelmingly modeled using the Poisson distribution, which is discrete. Thanks to the well-known approximation to the normal distribution, our assumption seems to be valid as long as both the demand rate and the no-show rate are at least 5 because the normal distribution is logconcave. Moreover, many commonly used distributions, such as uniform distribution, gamma distribution (with a shape parameter greater than 1), beta distribution, Weibull distribution, Laplace distribution, logistics distribution, exponential distribution, and extreme value distribution, are logconcave. Even if the airline company has only empirical data available, we would present the empirical distribution as the data following an underlying continuous distribution. By the Glivenko-Cantelli Theorem (see Van der Vaart [2000]), the empirical distribution estimates the cumulative distribution function and converges with a probability of 1. It is very likely that the underlying distribution would be approximated with the known logconcave distributions. Furthermore, the demands of multiple booking classes would be jointly distributed, i.e., the demand uncertainty of booking classes will be correlated rather than independent. This is an advantage of the solution technique for the chance-constrained optimization because it is closer to the reality. Computationally speaking, as long as we can generate a large enough correlated random data set as the input of simulation, the solution technique will guarantee a satisfactory solution, which is close to the true optimal (see Linderoth et al. [2006]). In our numerical results, we use correlated normal distribution data to feed simulations.
3 The cost of denying boarding to the lowest-fare customers To gain managerial implications regarding overbooking, we have two models to compare: one model of seat allocation with overbooking, and another of overbooking with denying boarding to customers. The difference between the two models occurs if the chance constraint, which is designed to assure seat availability for higher-paying customers, is in the model or not. We need the following results to reach our conclusions: Lemma 1. Both ν ∗ (λ) and νb∗ (λ) are monotonically increasing functions with respect to λ. We just need to show the case for ν ∗ (λ) of model (MoI). When we increase λ to a level at which neither of the constraints is binding, the optimal solution will not be affected by these constraints, regardless of the value of λ. When the constraints Ax ≤ λc, at least some of them, are binding, increasing λ means a larger feasible region and suggests a greater objective value. Thus, the optimal value ν ∗ (λ) will be monotonically increasing. A similar argument will apply to model (MoC). Lemma 2. Both ν ∗ (λ) and νb∗ (λ) are concave functions with respect to λ. 11
This is true for ν ∗ (λ) because (MoI) is a linear program, and λ is on the right-hand side of the constraints only. Similarly, it is true for model (MoC) because it is equivalent to a convex program (2.3), and λ is on the right-hand size of the constraints. Please refer to Boyd and Vandenberghe [2004] for more details. We omit the proof here. Lemma 3. For λ ≥ 1, we have ν ∗ (λ) − νb∗ (λ) ≥ 0. This result is also concluded by the feasible region argument. In model (MoC), we have the chance constraint, which will greatly narrow down the feasible region. Thus, the optimal objective value under a narrower feasible region would be nonincreasing for any λ. The results regarding the optimal solutions x∗ (λ) = [x∗H (λ); x∗E (λ)] and x∗b (λ) = [x∗Hb (λ); x∗Eb (λ)] are less straightforward, but more insightful. It is rather difficult to reach an analytical conclusion regarding the component-wise comparison between x∗ (λ) and x∗b (λ). Thanks to the fact that the airlines will deny boarding to only the lowest-fare customers, and the higher-paying customers are protected for each itinerary, we only need to compare n pairs of scalars, x∗E (λ) and x∗Eb (λ), by their components. For the lowest-fare customers, we have: Lemma 4. fET E[min{x∗Eb (λ), ξE − ζE }] ≤ fET E[min{x∗E (λ), ξE − ζE }] for any λ ≥ 1, where fE ∈ Rn is the fare levels for the lowest-fare customers of n itineraries. Proof. For the MoI model, when we maximize the total revenue, we allocate all λc to both higherpaid and lowest-fare customers, i.e. AH x∗H + AE x∗E = λc For the MoC model, we have an additional chance constraint being imposed to guarantee the seat availability of the higher-paid booking classes, P(AH (ξH − ζH ) ≤ [AH 0]x) ≥ 1 − α, AH x∗bH + AE x∗bE = λc which suggests AH x∗H ≤ AH x∗bH and AE x∗E ≥ AE x∗bE for any network structure A. In this paper, the customers are either higher-paid or the lowest-fare customer and AE is one-dimensional within each itinerary. That is, more seats will be allocated to the pool of higher-paying customers and fewer seats for the lowest-fare customers., i.e., fj x∗jb (λ) ≤ fj x∗j (λ), j ∈ E, which induces fET x∗Eb (λ) ≤ fET x∗E (λ). For any realized ξ and ζ, we have fET min{x∗Eb (λ), ξE − ζE } ≤ fET min{x∗E (λ), ξE − ζE }. By a similar argument, when airlines deny boarding to the lowest-fare customers, the expected revenue from the higher-paying customers are non-decreasing, as follows: 12
T E[min{x∗ (λ), ξ − ζ }] ≥ f T E[min{x∗ (λ), ξ − ζ }] for any λ ≥ 1, where f ∈ Rn Lemma 5. fH H H H H H H H Hb
is the fare levels for the higher-paying booking classes of n itineraries. Proof. Unlike the lowest-fare customers, there are multple booking classes which are higher-paid. Rather than separating these booking classes, we aggreate all of higher-paying booking classes as one class. Thus, we can still use the argument from the previous Lemma that the total number of seats, regardless of the booking classes as long as these classes are higher-paying, we have AH x∗H ≤ AH x∗bH for any AH . Lemma 3 shows that model (MoC) of denying boarding to customers would have a lower expected revenue, and Lemma 4 suggests that for the lowest-fare classes, the expected revenue of the higher-paying customers would be better when airlines deny boarding to customers. That is: T E[min{x∗Hb (λ), ξH −ζH }−min{x∗H (λ), ξH −ζH }] fET E[min{x∗E (λ), ξE −ζE }−min{x∗Eb (λ), ξE −ζE }] ≥ fH
Since overbooking is intended to assure seat availability for the higher-paying customers in extreme cases, the expected revenue of the higher-paying customers would not be significantly higher, even if the model assigns more seats to satisfy the chance constraint. For the lowest-fare customers, however, a lower number of seats means a significant reduction in the expected revenue. Such a reduction would be the impact of denying boarding to customers. For example, the regular overbooking model assigns x∗j (λ) seats to the lowest-fare customers, and model (MoC) assigns x∗jb (λ) for itinerary j. By Lemma 4, we have x∗j (λ) ≥ x∗jb (λ), and the seat allocation x∗j (λ) will be implemented. Thus, we conclude that fET E[min{x∗E (λ), ξE − ζE } − min{x∗Eb (λ), ξE − ζE }]
(3.1)
would be the expected impact of denying boarding to customers. For itinerary j, by the probability of 1 − α, there will be up to x∗j (λ) − x∗jb (λ) lowest-fare customers who will be denied boarding with a loss of sales at fj [x∗j (λ) − x∗jb (λ)]
(3.2)
The value of (3.2) provides answers to questions associated with overbooking.
3.1 The impact of denying boarding to customers First, if an airline has to deny boarding to one or more lowest-fare customers, the total compensation would be up to the value of (3.2). For the case of denying boarding to multiple customers, the 13
compensation package would be evenly split among recipients. Consider an airline that overbooks on itinerary j with λ ≥ 1. The optimal solutions of models (MoI) and (MoC) are x∗j (λ) = 30 and x∗jb (λ) = 10, respectively. It means that the airline will allocate 30 seats to the lowest-fare customers, and there will be up to x∗j (λ) − x∗jb (λ) = 20 lowest-fare customers who may be denied boarding under an extreme case of demand and no-shows. The fare price is fj = $200, and the airlines would determine whether overbooking will be profitable for them. Since there are federal regulations regarding overbooking, airlines would simply compensate the denied-boarding customers with the amount complying with the federal regulations. For example, there will be one denied-boarding customer who is entitled to a maximum of $1, 350 compensation. If the value of (3.2) is $4, 000, the overbooking operation would increase the revenue by $2, 650. However, if there are two deniedboarding customers who are entitled to $2, 700, the overbooking operation will profit the airline only by $1, 300. If there are 3 or more customers denied boarding, the airline will start losing profits. Using our model, the airline can decide whether to continue overbooking the flight. We note that our method would take λ as a predetermined value rather than as a major parameter. This decision is easy to explain to customers and airline crews. First, λ represents the overbooking rate. For most airlines, the practices of overbooking is standardized with a fixed λ. A constantly changing λ indicates frequent changes to the booking operation, which is not a usual practice. Second, by Lemmas 1 and 2, changing λ will not significantly increase the expected revenue because a greater λ will lead to fewer binding constraints of Ax ≤ λc. When all of the constraints of Ax ≤ λc become non-binding, there will be no effect at all by increasing λ. Thus, although λ seems to be important, we are more focused on other managerial implications.
3.2 The elite customer status The airlines have the option of additional revenue from the lowest-fare customers. The overbooking operation has received criticism from the public and the media over the years because there may be unhappy customers. The overbooking operation would generate additional revenue on the lowest-fare customers by offering elite customer status, which is a service to exempt the lowest-fare customers from being denied boarding. Many lowest-fare customers have the same tight schedules as other travelers and they want to arrive on time with minimum cost. Whenever there is a conflict, these customers would like to pay additional amount to gurarntee their seats. To avoid such an unpleasant event, the airlines may offer alternatives to the lowest-fare customers, such as the elite customer status program. After joining the program, customers would share their profiles of past travel history and airlines would be able to estimate the prospective profits if these customers are loyal. According to travel advisories, maintaining an elite status will greatly lower the chance of
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being denied boarding. The elite status program will greatly benefit airlines from the customer loyalties. Of course, the elite customer status program will cost airlines by rewarding customers with mileage credits. In this paper, we will evaluate the amount of additional revenue generated from the elite customer status for overbooking-related decision support. Consider the case that there will be k customers of every lowest-fare booking class who want to purchase the elite customer status service. We must remark that, in this paper, we only consider the additional revenue which is generated from the lowest-fare customers. Usually, there are multiple booking classes of each flight and their prices are different. However, only the lowest-fare customers are subject to denied boarding. The additional revenue generated is the direct consequence of overbooking because our chance-constrained model intentionally allocates more seats to higher-paid customers. Thus, in this research, there are only two booking classes which would make a difference, the lowest-fare class and all other higher-paid classes. To calculate the service charge for this elite customer status, we need the following model: νb∗ (λ, k) := max f T E[min{x, ξ − ζ}] subject to c ≤ Ax ≤ λc P(AH (ξH − ζH ) + k ≤ [AH 0E ]x) ≥ 1 − α
(3.3)
x≥0 where k is the vector of n with all of the components valued at k. The choice of k is solely at airlines’ discretion for two reasons. First, the number of lowest-fare customers is highly random and the airline has to determine when the booking process is close to its end. Second, airlines rank customers by their past records, which may also be subject to large variations.
In this model, we adjust the demand by assigning k seats of each itinerary to the
higher-paying customers, which means that for each itinerary, k lowest-fare customers would be regarded as the higher-paying customers. We have Proposition 1. For k ≥ 1, νb∗ (λ, k) ≤ νb∗ (λ).
(3.4)
We just need to show that the new chance constraint P(AH (ξH − ζH ) + k ≤ [AH 0E ]x) ≥ 1 − α further narrows the feasible set of x. It is easy to see that for any x that satisfies (3.3) with k ≥ 1 will always satisfy (3.3) with k = 0. However, it is not true for any x that satisfies (3.3) with k = 0 to satisfy the case of k > 0. 15
By Proposition 1, we have k elite customer status sold at price p(λ, k) for each flight. The choice of k is determined by the airline. For a network of m flights, we have νb∗ (λ, k) + mkp(λ, k) ≥ νb∗ (λ) p(λ, k) ≥
νb∗ (λ) − νb∗ (λ, k) mk
(3.5)
(3.5) is true because the expected revenue under overbooking is no less than the expected revenue without overbooking. In our model, the chance constraint will allocate less seats to the lowestfare customers and more loss of sales under normal booking would occur. The difference between both objective values would be the expected loss due to the compensations paid to deny boarding to customers. Thus, a revenue p(λ, k) would be collected from lowest-fare customers to, at least break even the compensation paid. However, it will be unlikely to have the value of p(λ, k) in a closedform format, given the nature of the model. Also, the solution technique for chance-constrained optimization requires simulation, and the obtained solution will be, in probability, close enough to the optimal solution. The seat inventory model (2.3) provides two major managerial implications. First, by solving both seat allocation models, with and without the chance constraint for the higher-paying customers, the airline company understands the impact of denying boarding to a certain number of customers on overbooked flights. If the airlines have to compensate the denied-boarding customers at the rate of the federal regulations, the airline will determine whether or not to overbook the flight in the first place. Second, our method provides decision support for the airlines to generate revenue from the lowest-fare customers. Previously, the lowest-fare customers usually generate the least margin for the airlines, and it is rather difficult to increase the margin. By calculating the value of (3.5), the airlines will determine which lowest-fare customers with the elite customer status would not be subject to denying boarding. All of these implications will be calculated numerically, and we demonstrate the numerical results in Section 4.
4 Numerical results In this section, we present the simulation-based numerical approach to solve (MoI) and (2.3). Consider an independent and identically distributed sample of size N regarding the demand ξ and noshow ζ. The generalized realizations will be {ξ i } and {ζ i }, i = 1, . . . , N . For a large enough N , the obtained optimal solution would be close enough to the true optimal solution (see Kleywegt et al. [2002]). In this section, we will use both {ξ i } and {ζ i }, i = 1, . . . , N to replace the random variables.
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For problem (MoI), the feasible region is a polyhedron, and the objective function is convex, but nonlinear. We need the following linear programming model with additional variables y i ≥ 0, y i ∈ Rnp : N 1 X i max f x−f y N x,y 1 ,...,y i ,...,y N T
T
(4.1)
i=1
subject to : c ≤ Ax ≤ λc x − yi ≤ ξi − ζ i x, y i ≥ 0, i = 1, . . . , N With the same realizations of random variables, y i represents the number of seats not sold, and the original minimum function in the objective of (MoI) would be implemented. When xj ≤ ξji − ζji , i.e., the demand is strong and no seat flies empty, we have yji = 0 for the booking class j. Likewise, when xj ≥ ξji − ζji , the revenue of the yji seats would be adjusted. For problem (2.3), we add the chance constraint into (4.1) and we have: max
x,y 1 ,...,y i ,...,y N
fT x − fT
N 1 X i y N i=1
subject to : c ≤ Ax ≤ λc log(1 − α) − log FH ([AH 0]x) ≤ 0
(4.2)
x − yi ≤ ξi − ζ i x, y i ≥ 0, i = 1, . . . , N Model (4.2) is a convex optimization model with the linear objective function, and all but one constraint is linear. As long as we can assess the gradient (or subgradient if the gradient does not exist), we would solve the problem. The method of calculating the gradient or the subgradient of the chance constraint is in Chen [2017], and we omit the technical details regarding this method in this paper. For problem (3.3), we determine the price for the elite customer status for the lowest-fare customers. The model is: max
x,y 1 ,...,y i ,...,y N
fT x − fT
N 1 X i y N i=1
subject to : c ≤ Ax ≤ λc log(1 − α) − log FH ([AH 0]x − k) ≤ 0 x − yi ≤ ξi − ζ i x, y i ≥ 0, i = 1, . . . , N 17
(4.3)
where k seats of each flight will be entitled to the elite customer status. We conduct a simulation study on a moderate size network, which has been previously used as a representative problem instance in Kunnumkal and Topaloglu [2011] and Erdelyi and Topaloglu [2010]. The purpose of this study is to present our method of evaluating the compensation by our optimization models. The numerical results by simulation will greatly enrich the literature and provide insightful views for practitioners. Consider an airline operating on a network with 1 hub and 8 airports in Figure 1. There are 16 flights of 137 seats, and there are n = 72 itineraries connect-
Figure 1: Structure of the airline network of 1 hub and 8 airports. ing any pair of airports or hub, among which 16 itineraries are nonstop trips from or to the hub. Since this paper concerns the impact of denying boarding to the lowest-fare customers, we only have two booking classes, the lowest-fare class and the higher-paying class in this case. Therefore, there will be 72 × 2 = 144 decision variables, and matrix A is 16 × 144. The fare levels, demand and no-show information are in Table 1. In this section, we also study the case for low-cost airlines. By Trip
Class
Fare
Demand Mean
Demand Std
No-show Low
No-show High
Direct
Lowest
$300
21
4.58
0
2
Direct
Higher-paying
$500
12
3.46
0
4
Connect
Lowest
$800
11
3.31
0
2
Connect
Higher-paying
$1,200
7
2.65
0
4
Table 1: Fare levels, demand, and no-show settings for the major airlines assuming the same demand and no-show rates, we change the regular fare levels of Table 1 from $300, $500, $800, and $1,200 to $100, $150, $250, and $300, respectively. The numerical experiment is completed on a workstation running Windows 10 and Matlab R2018a with 32G memory and i76990K CPU. The method in Chen [2017] will provide the gradient (or subgradient) of the chance constraint. We use the fmincon function of Matlab as our primary solver. 18
In this numerical experiment, we assume that the airline overbooks 15% seats, i.e., λ = 1.15, and the airline needs to make sure that by at least 95% of chance, the higher-paying customers will board the flights. We answer the following questions regarding the overbooking operation.
4.1 Low-cost airlines vs. major airlines on overbooking We present our numerical results regarding the different decisions of overbooking for low-cost and major airlines. We simulate the number of denied-boarding lowest-fare customers and the difference between the objectives of the seat allocation and the overbooking models to calculate the compensation of each denied-boarding customer. In Table 2, the compensation level of low-cost Airline
Revenue ($)
Revenue(CC) ($)
Deny Boarding Counts
Compensation ($)
Chance of overbooking
Gain by Overbooking ($)
Major
814833 ± 39474
641545 ± 64750
46.64 ± 17
3863 ± 1779
0
2513 ± 1779
Low-cost
237640 ± 10649
178367 ± 12910
45.23 ± 17
1362 ± 597
0.542
12 ± 597
Table 2: Simulation summary with 1,000 replications for the major and low-cost airlines airlines is significantly lower than that of the major airlines. The federal regulations require airlines to pay up to $1,350 for domestic passengers. If the calculated compensation is greater than $1,350, it indicates that the overbooking operation is both profitable and worth defending. For the major airlines, the compensation value is far greater than $1,350 while there is a chance of 0.542 for lowcost airlines with a much lower compensation value. Thus, our numerical results explain the fact that low-cost airlines are reluctant to overbook flights, and the major airlines hold the overbooking operation firmly. Likewise, our analysis will draw similar conclusions for the European and Asian markets because the compensations are all regulated progressively.
4.2 About the elite customer status Selling elite customer status to the lowest-fare customers will provide additional revenue to the airline. The airline would like to know the price of the elite customer status and the number of offerings. In this numerical experiment, we tried to sell from 1 to 5 elite customer status, at both low-cost airlines and the major airlines, and we summarize the results in Figures 2 and 3. The numerical results suggest that the number of denied-boarding customers in both types of airlines is rather stable, regardless of selling the elite customer status and the number of offerings. Since a elite customer status will assure the seat for one lowest-fare customer, such an adjustment to the optimal seat allocation will make the seat allocation suboptimal and will decrease the expected revenue for both types of airlines. Our calculation suggests that with more customers guaranteed seats and a mild decrease in the expected revenue, the compensation of each customer in our model will decrease, as well when airlines offer the elite customer status to the lowest-fare customers. 19
Figure 2: Selling elite customer status at low-cost airlines However, a lower level of compensation to each denied-boarding customer at low-cost airlines will yield more meaningful results because there is an increasing chance that the compensation will be less than the federally regulated amount of $1,350. Thus, if the elite customer status service is offered to low-cost airlines, the overbooking operation becomes less appealing, and the elite customer status may not be adopted by low-cost airlines at all because low-cost airlines may entirely terminate the overbooking operation.
Figure 3: Selling elite customer status at the major airlines For major airlines, although we also observe a similar declining trend of compensation for all cases of offering the elite customer status, the compensation level is still, almost surely, greater than
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$1,350, and the major airlines will continue overbook their flights. We also conclude that the major airlines may offer the elite customer status on a limited scale because if more lowest-fare customers are assured seats, the expected revenue will deviate more from the optimal. Consequently, the cost of each elite customer status will increase to a level that the lowest-fare customers cannot afford. In our experiment, when only one elite customer status is offered at a cost of $60 in addition to the fare previously paid, it seems likely that a lowest-fare customer on a tight schedule may purchase the card. However, when the airline offers two or more elite customer status, the cost of each card increases well above $200. Given the fact that the lowest-fare customers are paying $100-$150, it is rather difficult to sell these elite customer status.
5 Conclusion remarks In this research, we propose the chance-constrained optimization model to solve a problem related to airline overbooking: evaluating the impact of denying boarding to the lowest-fare customers when flights are overbooked. For an overbooked flight, the airlines will surely deny boarding to lowest-fare customers to assure seat availability for the higher-paying customers. To model such an operation, we build an optimization model by adding one chance constraint to assure seat availability for the higher-paying customers by a high probability, such as 95% or higher across the network. We then compare both models, the seat allocation model with overbooking and our model, regarding many questions of overbooking. We gain managerial implications from our chance-constrained model. Although overbooking has been a topic for decades, the research on determining the compensation for denied-boarding customers is still rare. Through comparing optimization models, we would be able to numerically calculate the cost of denying boarding to customers, which will provide the airlines with decision support on whether to continue overbooking. We therefore can explain the reason that low-cost airlines are increasingly reluctant to overbook flights, while major airlines want their overbooking in place. We also would generate additional revenue by selling the elite customer status to some lowest-fare customers. We find that the revenue of selling elite customer status, on a limited basis, would ease the impact of denying boarding to customers and would make overbooking more profitable for the major airlines.
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