ACE is High 339 ACE is High Anthony C Pecorella Elizabeth a perez Crystal T. Taylor Wake Forest University Winston Fig5-Salem, Nc Advisor: Edward e. allen Introduction We design a model that allows an airline to substitute its own values for ticket prices, no-show rates and fees, compensation for bumped passengers and capacities to determine its optimal overbooking level Our model is based on an equation that combines the two cases involved in overbooking: The first sums all cases in which the airline doesn't fill all seats with passengers, and the second sums all cases in which there is an over- flow of passengers due to overbooking. The model includes the possibility of upgrading passengers from coach to first-class when there is overflow in coach Furthermore, we use a binomial distribution of the probabilities of bumping booking percenta oly the airlines with useful information pertaining to customer relations We apply our model with different values of the parameters to determine of byusing our model, an individual airline can find an optimal overbooking timal overbooking levels in different situations level that maximizes its revenue. an joint optimal overbooking strategy for all airlines is to agree to allow bumped passengers to fly at a discounted fare on a Analysis of the Problem From January to September 2001, 0. 187% of pa bumped from flights due to overbooking. This seems like an inconsequential percentage, but it actually amounts to 731, 449 people. Additionally, 4.4% of those bumped, The UMAP Journal 339-349. @Copyright 2002 by COMAP, Inc. All rights reserved ermission to make digital or hard copies of part or all of this work for personal or classroom m use is granted without fee provided that copies are not made or distributed for profit or commercial dvantage and that copies bear this notice. Abstracting with credit is permitted but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP
ACE is High 339 ACE is High Anthony C. Pecorella Elizabeth A. Perez Crystal T. Taylor Wake Forest University WinstonFig5-Salem, NC Advisor: Edward E. Allen Introduction We design a model that allows an airline to substitute its own values for ticket prices, no-show rates and fees, compensation for bumped passengers, and capacities to determine its optimal overbooking level. Our model is based on an equation that combines the two cases involved in overbooking: The first sums all cases in which the airline doesn’t fill all seats with passengers, and the second sums all cases in which there is an over- flow of passengers due to overbooking. The model includes the possibility of upgrading passengers from coach to first-class when there is overflow in coach. Furthermore, we use a binomial distribution of the probabilities of bumping passengers, given different overbooking percentages, to supply the airlines with useful information pertaining to customer relations. We apply our model with different values of the parameters to determine optimal overbooking levels in different situations. By using our model, an individual airline can find an optimal overbooking level that maximizes its revenue. An joint optimal overbooking strategy for all airlines is to agree to allow bumped passengers to fly at a discounted fare on a different airline. Analysis of the Problem From January to September 2001, 0.187% of passengers were bumped from flights due to overbooking. This seems like an inconsequential percentage, but it actually amounts to 731,449 people. Additionally, 4.4% of those bumped, The UMAP Journal 339–349. c Copyright 2002 by COMAP, Inc. All rights reserved. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP
340 The UMAP Journal 23.3(2002) or 32, 452 people, were denied their flights involuntarily [U.S. Department of Transportation 2002 to 15% of passengers who reserve a seat don' t show up, airlines have little chance to fill their planes if they book only as many passengers as seats available. Overbooking by american Airlines helped save the airline $1. 4 billion between 1989 and 1992 We examine a fictional company to determine an optimal overbooking strat- egy that maximizes revenue. The goal is a model to increase revenue while maintaining favorable customer relations Our main model, the Expected Gain Model, provides a clear formula for what percentage of the seats to overbook. Based on sample no-show rates, ticket prices, and seat numbers our Expected Gain Model shows that a 16% overbooking rate is the most effective choice Our other model, the Binomial distribution Model, calculates for various overbooking levels. the probability that a passenger will be bumped Assumptions There isno overbooking in first class (to maintain good relations with wealthy and influential passengers Anyone bumped (voluntarily or involuntarily)is compensated with refund of ticket price plus an additional 100% of the ticket price There are only two flight classes, coach and first-class The fare is constant regardless of how far in advance the ticket is purchased Overbooked passengers are given seats on a first-come-first-served basis, as is often the case. Therefore, ticket prices will average out for both those bumped and those seated Each passenger's likelihood of showing up is independent of every other passenger First-class ticketholders have unrestricted tickets which allow a full refund n case of no-show; coach passengers have restricted tickets, which allow only a 75% refund in case of no-show ● There are no walk-ons There are no flight delays or cancellations The marginal cost of adding a passenger to the plane is negligible
340 The UMAP Journal 23.3 (2002) or 32,452 people, were denied their flights involuntarily [U.S. Department of Transportation 2002]. Since 10% to 15% of passengers who reserve a seat don’t show up, airlines have little chance to fill their planes if they book only as many passengers as seats available. Overbooking by American Airlines helped save the airline $1.4 billion between 1989 and 1992. We examine a fictional company to determine an optimal overbooking strategy that maximizes revenue. The goal is a model to increase revenue while maintaining favorable customer relations. Our main model, the Expected Gain Model, provides a clear formula for what percentage of the seats to overbook. Based on sample no-show rates, ticket prices, and seat numbers, our Expected Gain Model shows that a 16% overbooking rate is the most effective choice. Our other model, the Binomial Distribution Model, calculates, for various overbooking levels. the probability that a passenger will be bumped. Assumptions • There is no overbooking infirst class (to maintain good relations with wealthy and influential passengers). • Anyone bumped (voluntarily or involuntarily) is compensated with refund of ticket price plus an additional 100% of the ticket price. • There are only two flight classes, coach and first-class. • The fare is constant regardless of how far in advance the ticket is purchased. Overbooked passengers are given seats on a first-come-first-served basis, as is often the case. Therefore, ticket prices will average out for both those bumped and those seated. • Each passenger’s likelihood of showing up is independent of every other passenger. • First-class ticketholders have unrestricted tickets, which allow a full refund in case of no-show; coach passengers have restricted tickets, which allow only a 75% refund in case of no-show. • There are no walk-ons. • There are no flight delays or cancellations. • The marginal cost of adding a passenger to the plane is negligible
ACE is High 341 The model Equations prob(a, y, r) r3(1-r)2 P1(g)=> prob(Sf, h, R/)[G-B)((y-(S; -k)+ Fc(S,-k) k=Sf+l-y P prob(Sf, k, Rf)Fc Mi(ac)=> prob(a, i, Re)(Fci+N(r-i)) M2(ax)=2 prob(r,j, Re)[ScFc+Nc(a-j)+PGj-Sc)+P2(j-se) j=Sc+1 M(x)=M1(x)+M2(x) Parameters Sf= seating available for first-class Sc seating available for coach R=show-up rate for first-class reservations Rc = show-up rate for coach reservations F= coach fare N. =no-show fee for coach B =coach bump cost to airline Variables .r= number of reservations Functions M()=expected gain with reservations prob(a, y, r)= probability of y events happening in a trials where r is the chance of a single event happenir Pill, Ply: to be described later
ACE is High 341 The Model Equations prob(x, y, r) = x y ry(1 − r) x−y P1(y) = Sf k=Sf +1−y prob(Sf ,k,Rf ) (−Bc) (y − (Sf − k) + Fc(Sf − k) P2(y) = S f −y k=0 prob(Sf ,k,Rf )Fcy M1(x) = S f −y i=0 prob(x, i, Rc) FC i + Nc(x − i) M2(x) = x j=Sc+1 prob(x,j,Rc) ScFc + Nc(x − j) + P1(j − Sc) + P2(j − Sc) M(x) = M1(x) + M2(x) Parameters Sf = seating available for first-class Sc = seating available for coach Rf = show-up rate for first-class reservations Rc = show-up rate for coach reservations Fc = coach fare Nc = no-show fee for coach Bc = coach bump cost to airline Variables • x = number of reservations Functions • M(x) = expected gain with x reservations • prob(x, y, r) = probability of y events happening in x trials where r is the chance of a single event happening • P1[y], P2[y]: to be described later
342 The UMAP Journal 23.3(2002) Binomial distribution model We create ACE Airlines, a fictional firm to understand better how to handle overbooking. We examine binomial distributions of ticket sales, so we call this the binomial distribution model ACE features planes with 20 first-class seats and 100 coach seats. The no- show rate is 10% for coach and 20% for first-class. Figure 1 compares various overbooking levels with the chance that there will be enough available seats in first-class to accommodate the overflow. The functions are ∑()0.9y(0(oach y=∑(2)08y(0)0)( irst class) where C reservations are made for coach and 20 are always made for first class Figure 1. Probability of enough seats vs overbooking level. The graph on the right is a close-up of the upper left corner of the one on the left Where the first-class line passes below the various overbooking lines indi- cates the probability at which we must start bumping passengers This simplistic model doesnt account for ticket prices, no-show fees, or re- funds to bumped passengers and doesn't specifically deal with revenue either Thus, it can act as a good reference for verifying the customer-relations aspect of any solution but can t give a good solution on its own. To be sure that aCe the most revenue it can we must create a more in-depth model aCe coach fare is $200. We refund $150 on no-show coach tickets thus gaining $50 on each. To keep good customer relations, when we are forced to bump a passenger from a flight, we refund the ticket price with an additional bonus of 100% of the ticket price(thus, we suffer a $200 loss) happening in c trials, with a probability r of each event happel.ent events Ne define prob(, y, r)as the binomial probability of y independe )(a-r)
0 .0000% 1 0.0000% 2 0.0000% 3 0.0000% 4 0.0000% 5 0.0000% 6 0.0000% 7 0.0000% 8 0.0000% 9 0.0000% 100.0000% 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 First Class 101 105 106 107 108 109 110 111 112 113 114 115 116 8 5.0000% 8 7.0000% 8 9.0000% 9 1.0000% 9 3.0000% 9 5.0000% 9 7.0000% 9 9.0000% 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 First Class 101 105 106 107 108 109 110 342 The UMAP Journal 23.3 (2002) Binomial Distribution Model We create ACE Airlines, a fictional firm, to understand better how to handle overbooking. We examine binomial distributions of ticket sales, so we call this the Binomial Distribution Model. ACE features planes with 20 first-class seats and 100 coach seats. The noshow rate is 10% for coach and 20% for first-class. Figure 1 compares various overbooking levels with the chance that there will be enough available seats in first-class to accommodate the overflow. The functions are y = 100+ x j=0 C j (0.9)j (0.1)C−j (coach), y = 20 −x j=0 C j (0.8)j (0.2)20−j (first class), where C reservations are made for coach and 20 are always made for first class. Figure 1. Probability of enough seats vs. overbooking level. The graph on the right is a close-up of the upper left corner of the one on the left. Where the first-class line passes below the various overbooking lines indicates the probability at which we must start bumping passengers. This simplistic model doesn’t account for ticket prices, no-show fees, or refunds to bumped passengers and doesn’t specifically deal with revenue either. Thus, it can act as a good reference for verifying the customer-relations aspect of any solution but can’t give a good solution on its own. To be sure that ACE is receiving the most revenue it can, we must create a more in-depth model. ACE coach fare is $200. We refund $150 on no-show coach tickets, thus gaining $50 on each. To keep good customer relations, when we are forced to bump a passenger from a flight, we refund the ticket price with an additional bonus of 100% of the ticket price (thus, we suffer a $200 loss). We define prob(x, y, r) as the binomial probability of y independent events happening in x trials, with a probability r of each event happening: prob(x, y, r) = x y ry(1 − r) x−y.
ACE is High 343 Model for coach We first ignore first class and maximize profit based solely on overbooking the coach section via the Simple expected gain Model. This model is defined in two parts. The first looks at the chances of the cabin not filling-i< 100 people showing up. Ace gets $200 for each of the i passengers who arrive and fly and $50 from each of the(a-i)no-shows. We multiply the probability of each outcome(determined by the binomial distribution) by the resulting revenue and sum over all of these values of i to find the M1(x)=∑prob(x,1,0.9(200150(x-2) The second part of the model focuses on overflow in the coach section, when 3>100. In this case, ACE is limited to $200 fare revenue on 100 passengers, plus $50 for each of the(a-i)no-shows. However, for the(j-100) passengers who arrive but have no seats, ACE bumps them and thus loses $200 in compensation per passenger. We again multiply by the probability of each outcome and sum M2(x)=∑prob(x,,0.9)2000150(x-1)-200-100小 We add Mi and ma to arrive at an expression M forrevenue. From the graph for M, we discover that (independent of first class)for maximum revenue, ACE should overbook by about 1l people, expecting a net revenue from the coach section of $20,055 (Figure 2) Simple Expected Gain Mode Revenue($ 20000 Coach reservations made igure 2. Simple Expected Gain Model: Revenue M vS number of coach reservations Howeve Coach Plus first Class When we add in consideration of first-class openings, ACE can overbook by even more while still increasing revenue, since it can upgrade coach overflow
ACE is High 343 Model for Coach We first ignore first class and maximize profit based solely on overbooking the coach section, via the Simple Expected Gain Model. This model is defined in two parts. The first looks at the chances of the cabin not filling—i < 100 people showing up. ACE gets $200 for each of the i passengers who arrive and fly and $50 from each of the (x − i) no-shows. We multiply the probability of each outcome (determined by the binomial distribution) by the resulting revenue and sum over all of these values of i to find the expected gain, M1: M1(x) = 100 i=0 prob(x, i, 0.9) 200i + 50(x − i) . The second part of the model focuses on overflow in the coach section, when j > 100. In this case, ACE is limited to $200 fare revenue on 100 passengers, plus $50 for each of the (x−i) no-shows. However, for the (j −100) passengers who arrive but have no seats, ACE bumps them and thus loses $200 in compensation per passenger. We again multiply by the probability of each outcome and sum: M2(x) = x j=101 prob(x,j, 0.9) 200(100) + 50(x − j) − 200(j − 100) . We addM1 andM2 to arrive at an expressionM for revenue. From the graph for M, we discover that (independent of first class) for maximum revenue, ACE should overbook by about 11 people, expecting a net revenue from the coach section of $20,055 (Figure 2). Figure 2. Simple Expected Gain Model: Revenue M vs. number of coach reservations. Howeve Coach Plus First Class When we add in consideration of first-class openings, ACE can overbook by even more while still increasing revenue, since it can upgrade coach overflow