Airline Yield Management with Overbooking, Cancellations, and No-Shows JANAKIRAM SUBRAMANIAN

Transcription

1 Airline Yield Manageent with Overbooking, Cancellations, and No-Shows JANAKIRAM SUBRAMANIAN Integral Developent Corporation, 301 University Avenue, Suite 200, Palo Alto, California SHALER STIDHAM JR. Departent of Operations Research, CB 3180, Sith Building, University of North Carolina, Chapel Hill, North Carolina CONRAD J. LAUTENBACHER NationsBank, 100 N. Tryon St., NC , Charlotte, North Carolina We forulate and analyze a Markov decision process (dynaic prograing) odel for airline seat allocation (yield anageent) on a single-leg flight with ultiple fare classes. Unlike previous odels, we allow cancellation, no-shows, and overbooking. Additionally, we ake no assuptions on the arrival patterns for the various fare classes. Our odel is also applicable to other probles of revenue anageent with perishable coodities, such as arise in the hotel and cruise industries. We show how to solve the proble exactly using dynaic prograing. Under realistic conditions, we deonstrate that an optial booking policy is characterized by state- and tie-dependent booking liits for each fare class. Our approach exploits the equivalence to a proble in the optial control of adission to a queueing syste, which has been well studied in the queueing-control literature. Techniques for efficient ipleentation of the optial policy and nuerical exaples are also given. In contrast to previous odels, we show that 1) the booking liits need not be onotonic in the tie reaining until departure; 2) it ay be optial to accept a lower-fare class and siultaneously reject a higher-fare class because of differing cancellation refunds, so that the optial booking liits ay not always be nested according to fare class; and 3) with the possibility of cancellations, an optial policy depends on both the total capacity and the capacity reaining. Our nuerical exaples show that revenue gains of up to 9% are possible with our odel, copared with an equivalent odel oitting the effects of cancellations and no-shows. We also deonstrate the coputational feasibility of our approach using data fro a real-life airline application. In the airline industry, the practice of selling identical seats for different prices to axiize revenues is coonly referred to as yield anageent (YM) (or seat inventory control). Yield anageent is an exaple of a ore general practice known as revenue anageent (RM) or perishable inventory control, in which a coodity or service (such as the use of a hotel roo on a particular date) is priced differently depending on various restrictions on booking (e.g., advance purchase requireents) or 147 cancellation (e.g., nonrefundability or partial refundability). The coon thread in all these exaples is the perishability of the coodity: a seat on a particular flight is worthless after the flight departs, just as an unoccupied hotel roo generates no revenue. The YM proble studied in this paper can be described in its ost general for as follows. Consider a single-leg flight on an airplane of known capacity C. Passengers can belong to one of fare Transportation Science / 99 / $01.25 Vol. 33, No. 2, May Institute for Operations Research and the Manageent Sciences

2 148 / J. SUBRAMANIAN, S. STIDHAM, JR., AND LAUTENBACHER classes, class 1 corresponding to the highest fare and class to the lowest. Booking requests in each fare class arrive according to a tie-dependent process. Based on the nuber of seats already booked, we ust decide whether to accept or reject each request. Passengers who have already booked ay cancel (with known tie- and class-dependent probability) at any tie up to the departure of the flight. At the tie of cancellation, the passenger is refunded an aount that ay be tie and class dependent. Passengers can also be no-shows at the tie of departure. No-shows are refunded an aount that ay be different fro the aount refunded for cancellation. Overbooking is allowed, with corresponding penalties deterined by an overbooking penalty-cost function. Although, for concreteness, we use the terinology of airline yield anageent in this paper, any of our results are also applicable to other probles of RM. For exaple, in the hotel industry, roos are sold at different rates, depending on the tie until arrival and/or cancellation restrictions. Refunds for cancellation ay also depend on the rate. Our basic approach is to exploit the equivalence of this YM proble to a proble in the optial control of arrivals to a queueing syste, which has been the subject of an extensive literature over the past twenty-five years (see JOHANSEN and STIDHAM (1980), STIDHAM (1984, 1985, 1988), and STIDHAM and WE- BER (1993) for surveys). In this way, we are able to solve a substantially ore versatile and realistic odel than those previously studied in the literature on airline seat allocation. Aong the references ost relevant to this paper are (in chronological order) LITTLEWOOD (1972), BE- LOBABA (1987, 1989), CURRY (1990), WOLLMER (1992), BRUMELLE and MCGILL (1993), ROBINSON (1995), and LEE and HERSH (1993), all of who consider the single-leg proble. Belobaba (1987) provides a coprehensive overview of the seat inventory control proble and the issues involved. See Belobaba (1989), Lee and Hersh (1993), and LAUT- ENBACHER and STIDHAM (1996) for surveys and additional references. SMITH, LEIMKUHLER, and DAR- ROW (1992) review yield anageent practices as they are ipleented at Aerican Airlines. WEATHERFORD and BODILY (1992) (see also WEATHERFORD, BODILY, and PFEIFER (1993)) provide a useful taxonoy of perishable asset anageent (PARM) probles, which include YM and related probles. (In their terinology, our proble is (A1-B1-C1-D1-EI-F3-G3-H3-I1-J1-K1-L5-M2-N3).) WILLIAMSON (1992) and TALLURI and VAN RYZIN (1996) study the ulti-leg proble (network RM), which we do not consider in this paper. Other related papers are ALSTRUP et al (1986), CHATWIN (1992), GALLEGO and VAN RYZIN (1994, 1997), and YOUNG and VAN SLYKE (1994). A coon assuption in any of the earlier papers (e.g., Belobaba (1987), Belobaba (1989), Woller (1992), and Bruelle and McGill (1993)) has been that all requests for a fare class arrive earlier than the requests for higher fare classes. This assuption is known to be unrealistic. Robinson (1995) relaxes this assuption, but still assues that different fare classes book during nonoverlapping tie intervals. Lee and Hersh (1993) consider the seat inventory control proble without aking any assuption on the arrival pattern. They use dynaic prograing ethodology to solve for the optial policy. They do not, however, perit cancellations, no-shows, or overbooking. The rest of this paper is organized as follows. We begin in Section 1 with a siple extension of the odel of Lee and Hersh (1993) that incorporates cancellations, no-shows, and overbooking (Model 1). The cancellation and no-show probabilities, although allowed to be tie dependent, are assued to be the sae for all classes. There are no refunds for cancellations or no-shows, but the fare paid in each class ay be tie dependent. (Later, in Section 2, we show how to accoodate refunds by an equivalent-charging schee borrowed fro queueing-control theory, which results in a net fare that is tie dependent, even if the gross fare is not.) As in Lee and Hersh (1993), the planning horizon is divided into discrete tie periods, in each of which at ost one event (reservation request or cancellation) can occur. We forulate the proble as a finitehorizon, discrete-tie Markov decision process (MDP) in which the state variable is the total nuber of seats already booked. With booking requests viewed as custoer arrivals and cancellations as service copletions, the proble becoes equivalent to a proble in the optial control of arrivals to a queueing syste with an infinite nuber of servers. Exploiting this equivalence, we show that the YM proble satisfies well-known sufficient conditions for an optial policy to be onotonic, which, in this context, translates to the existence of tie-dependent booking liits for each class. Previously, the optiality of such a policy has been established in the YM literature only for the proble without cancellations or no-shows, and then by coplicated arguents apparently dependent on the particular structure of the YM proble. In Section 2, we extend the basic odel to allow class-dependent cancellation and no-show probabilities and refund aounts, with refunds at the tie of cancellation or no-show (Model 2). In this case, the

3 OVERBOOKING, CANCELLATIONS, AND NO-SHOWS / 149 forulation as an MDP requires a ultidiensional state variable, because now one ust know the nuber of seats reserved in each fare class to predict future cancellations and no-shows (because the rates at which these occur are now allowed to be class dependent). The curse of diensionality precludes solving probles of ore than odest size, especially considering that an airline is faced with solving such probles for hundreds of legs on a daily basis. In the special case, in which cancellation and no-show probabilities are not class dependent, however, we show how to transfor the proble into an equivalent MDP in which the expected lost revenue fro cancellation or no-show refunds is subtracted fro the fare paid at the tie of booking. (This is the equivalent charging schee referred to above.) This equivalent proble satisfies the conditions of Model 1, for which the total nuber of seats currently booked is a sufficient state description, thus reducing the proble to a one-diensional MDP and avoiding the curse of diensionality. The assuption that cancellation and no-show probabilities are class independent is not realistic, but still represents an iproveent on previous odels, which ignore the effects of cancellations and no-shows altogether. (In Appendix B, we propose a heuristic approxiation for the case of class-dependent cancellation and no-show probabilities which retains the one-diensional state variable and, thus, greatly enlarges the space of solvable probles.) Section 3 contains our nuerical results. First, we apply Model 1 to several single-leg YM probles with cancellations, no-shows, and overbooking, in which the (tie-dependent) cancellation and noshow probabilities are the sae for all classes. Our nuerical results deonstrate that the optial booking liits ay not be nested by fare class, in contrast to the situation without cancellation or overbooking. More accurately, the order in which the classes are nested ay be tie dependent, based on the net fare: the gross fare inus the expected cancellation and/or no-show refund. We also copare our results to those for dynaic-prograing odels that do not allow for the possibility of cancellation and/or no-shows, such as the odel of Lee and Hersh (1993). Even if one refines such odels by subtracting the expected lost revenue caused by cancellations and no-shows fro the fare, these odels do not include the cancellation and no-show probabilities in their calculations of future state-transition probabilities. We show, by eans of nuerical exaples, that our odel can yield substantial increases in revenues up to 9% when copared to an equivalent odel oitting the effects of cancellations and no-shows. We also deonstrate the coputational feasibility of our odel, using data fro a real-life airline application with six fare classes and a 331-day booking horizon for a plane with 100 seats. Next, we consider Model 2, with class-dependent cancellation and no-show probabilities. We solve a sall but realistic odel, using the ultidiensional MDP forulation discussed in Section 2. Then, we copare this (exact) solution to one-diensional approxiations. In Appendix A, we show how to apply our results to solve the continuous-tie seat-allocation proble, in which requests for seats in each fare class arrive according to (tie-dependent) Poisson processes (cf. Lee and Hersh, 1993). We give two (equivalent) approaches, one based on uniforization (LIPP- MAN, 1975; SERFOZO, 1979) and the other on a tie discretization (Lee and Hersh, 1993). Appendix B presents our heuristic approxiation for the case of class-dependent cancellation and noshow probabilities. 1. BASIC DISCRETE-TIME MODEL (MODEL 1) IN THIS SECTION, we introduce and analyze our first discrete-tie odel for airline seat allocation (Model 1), which generalizes the single-seat odel of Lee and Hersh (1993). There are fare classes and N decision periods or stages, nubered in reverse chronological order, n N, N 1,..., 1, 0, with stage N corresponding to the opening of the flight for reservations and stage 0 corresponding to its departure. At each stage, we assue that one (and only one) of the following events occurs: (1) an arrival of a custoer (i.e., a request for a seat) in fare class i, i 1,..., ; (2) a cancellation by a custoer currently holding a reservation; or (3) a null event (represented as the arrival of a custoer of class 0). (In Appendix A, we show how this assuption is satisfied naturally when a continuous-tie proble is approxiated either by uniforization or tiediscretization.) In this odel, cancellations and no-shows occur at class-independent rates, which allows us to use a one-diensional state variable. Later (in Section 2) we shall relax this assuption. Let p in denote the probability of a request for a seat in fare class i in period n. Siilarly, let q n (x) and p 0n (x) denote the probability of a cancellation and a null event, respectively, in period n, given that the current nuber of reserved seats equals x. We assue that q n (x) is a nondecreasing and concave function of x, for each n. By our assuption that, at ost, one request or cancellation can occur at each

4 150 / J. SUBRAMANIAN, S. STIDHAM, JR., AND LAUTENBACHER stage, we have p in q n x p 0n x 1, (1) for all x and n 1. If the event occurring at stage n is the arrival of a seat request, the syste controller (the booking agent) decides whether or not to accept the request, based on the fare class i of the custoer and the current state x. If the event is a cancellation or null event, then no decision is ade. If the booking agent accepts a request for a seat in fare class i at stage n, the airline earns revenue r in 0. (By allowing the revenue to depend on n, we shall be able, in subsequent analysis, to incorporate the effects of cancellation and no-shows by the equivalent charging schee referred to in the Introduction.) There are also no-shows. We assue that, at the tie of departure, each custoer holding a seat reservation is a no-show with probability. Let Y(x) denote the nuber of people who show up for the flight, given that the nuber of reserved seats is x just before departure, so that x Y(x) is the nuber of no-shows. Because each custoer has a probability 1 of showing up for the flight, it is clear that Y(x) has a binoial-(x, 1 ) distribution. Let C denote the capacity of the airplane. If, at the tie of the departure, Y(x) y, then we incur an overbooking penalty (y). We assue that (y) is non-negative, convex, and nondecreasing in y 0, with (y) 0 for y C. REMARK 1. Note that we are assuing that requests for seats are independent of the nuber of seats already booked (a realistic assuption), whereas cancellation and no-show probabilities depend on the total nuber of booked seats, but not on the nuber booked in each class. The assuption that q n (x) is nondecreasing in x expresses the plausible property that the higher the nuber of seats already booked, the higher the probability of a cancellation. The concavity assuption is needed for technical reasons (in the proof that a booking-liit policy is optial). It is satisfied in all the applications discussed in this paper. It holds, for exaple, when q n (x) q n x, as is the case when the discrete-tie odel is derived fro a continuous-tie odel with a nonhoogeneous Poisson arrival process (see Appendix A). REMARK 2. The assuption that (y) is convex and nondecreasing is realistic. For exaple, (y) could be piecewise linear, k y a j y b j, j 1 where C b 1 b 2... b k and a j 0, j 1,..., k. In this case, the airline ust pay a 1 to each of the first b 2 b 1 passengers who volunteer to take a later flight, a 1 a 2 to each of the next b 3 b 2, and so forth. Our objective is to axiize the expected total net benefit 1 of operating the syste over the horizon fro period N to period 0, starting fro state x 0, that is, with no seats booked, at the beginning of period N. The proble can be forulated as a discrete-tie MDP with stages n N, N 1,...,0, in which the state x is the current nuber of booked seats. Because our odel allows for the possibility of overbooking, cancellations, and no-shows, the state variable need not satisfy the constraint x C, aswe have noted above. However, because we start with no seats booked at stage N and, at ost, one seat request can be accepted at each stage (because, at ost, one arrives), it follows that at each stage n, x N n. (To reduce the coputational burden in applications, one ay wish to introduce an additional constraint, x C v, where v is the overbooking pad: the axial aount by which the airline is willing to overbook. See Reark 4 below.) As a function of the state x in period n, let U n (x) denote the axial expected net benefit of operating the syste over periods n to 0. The optial value functions, U n, are deterined recursively by U n x p in ax r in U n 1 x 1, U n 1 x q n x U n 1 x 1 p 0n x U n 1 x, 0 x N n, n 1, (2) U 0 x E Y x, 0 x N, (3) where Y(x) Bin(x, 1 ). Now we show how to write this optiality equation in an equivalent for that is characteristic of a proble in the optial control of adission to a queueing syste. Let p n : p in, and let R n be a rando variable with probability ass function P{R n r in } 1 An alternative objective function is the expected total discounted net benefit. The analysis of this proble is substantially the sae. See JANAKIRAM, STIDHAM, and SHAYKEVICH (1994).

5 OVERBOOKING, CANCELLATIONS, AND NO-SHOWS / 151 p in /p n, i 1,...,, n 1. Let V n x, r : ax r U n 1 x 1, U n 1 x, for all x 0, 1,..., N n and real nubers r. (Here, r is a realization of the rando variable R n. In the present proble, the only values of r that have positive probability are of the for r r in for soe i 1,...,. In ore general queueingcontrol odels, the distribution of R n ay be arbitrary, with support [0, ).) Then, substituting in Eq. 2 and using Eq. 1, we obtain the equivalent optiality equations U n x p n E V n x, R n q n x U n 1 x 1 1 p n q n x U n 1 x, 0 x N n, n 1 (4) V n x, r ax r U n 1 x U n 1 x 1, 0 U n 1 x, 0 x N n, n 1 (5) U 0 x E Y x, 0 x N. (6) In this for, the optiality equations can be seen to be forally equivalent to those for optial control of adission to a queueing syste over a finite horizon as studied, for exaple, in LIPPMAN and STIDHAM (1977) (see also STIDHAM, 1978; Johansen and Stidha, 1980; HELM and WALDMANN, 1984; Stidha, 1984, 1985, 1988). Lippan and Stidha (1977) study a queue with a Poisson arrival process and a state-dependent exponential service echanis, which, after uniforization, reduces to a discrete-paraeter MDP whose finite-horizon optiality equations (see Eqs. 2 and 3 in Lippan and Stidha, 1977) have exactly the sae structure as our optiality Eqs. 4 and 5. (In Appendix A, we shall discuss how to use uniforization to apply our odel to a continuous-tie seat-allocation proble in which seat requests in each fare class arrive according to a Poisson process, and cancellations are governed by exponential distributions.) The odel of Lippan and Stidha (1977) accoodates a holding cost, which is a convex, nondecreasing function of the state. In our odel, the holding cost is identically zero. Lippan and Stidha (1977) use induction on the reaining nuber of stages, n, to prove that the optial value function, U n, is concave and nonincreasing, fro which it follows that an optial adission policy is onotonic in the state. (Although their odel assues that rewards, arrival rates, and service rates do not depend on n, the inductive proof is not affected by peritting dependence on n, as we do in our seat-allocation odel. See Johansen and Stidha (1980) and Hel and Waldann (1984), for exaple, for related odels allowing tie dependence.) To apply their results to our odel, we first need to verify that the function, U 0 (x) E[ (Y(x))], is concave and nonincreasing in x to start the induction. We use the following result fro the theory of stochastic ordering. (See Exaple 6.A.2 in SHAKED and SHANTHIKUMAR, 1994.) LEMMA 1. Let f(y), y 0, be a nondecreasing convex function. For each non-negative integer x, let Y(x) be a binoial-(x, ) rando variable (0 1) and let h(x) : E[f(Y(x))]. Then h(x) is nondecreasing convex in x {0, 1,...}. Because is convex and nondecreasing by assuption, it follows fro Lea 1 and Eq. 6 that U 0 is concave and nonincreasing. Then, by induction on n (cf. Theore 1 in Lippan and Stidha, 1977), U n is concave and nonincreasing, so that U n (x) U n (x 1) is nondecreasing in x 0, 1,...,N n 1. The difference, U n (x) U n (x 1), is the opportunity cost of accepting a booking at stage n 1, that is, the expected loss in future revenue that would result fro accepting the booking. In our odel, this opportunity cost plays the sae role as the expected arginal seat revenue (EMSR) in the pioneering work of Belobaba (1989) (see also Bruelle and McGill, 1993; Woller, 1992). It is also the optial bid price for our singleleg proble, in the sense of Williason (1992) (see also Talluri and van Ryzin, 1996). For each stage n and each fare class i, define the booking liit b in as b in : in x: U n 1 x U n 1 x 1 r in. (7) (If U n 1 (x) U n 1 (x 1) r in for all x, set b in.) Because U n 1 (x) U n 1 (x 1) is nondecreasing in x, b in is well defined, and an optial policy will have the for accept a fare-class-i request in state x at stage n if and only if 0 x b in. In the notation of Lee and Hersh (1993), b in C ŝ i (n) 1, where ŝ i (n) is the critical booking capacity for class i. An optial policy will therefore accept a fare-class i request at stage n if and only if s ŝ i (n), where s C x is the nuber of seats still available (the booking capacity). REMARK 3. In contrast to Lee and Hersh (1993), we allow the fares r in to depend on n as well as i, and we do not assue that the ordering of fares by class

6 152 / J. SUBRAMANIAN, S. STIDHAM, JR., AND LAUTENBACHER is the sae for all stages n. So, there ay be two classes i and j and stages n and k such that whereas r in U n 1 x U n 1 x 1 r jn, r ik U k 1 x U k 1 x 1 r jk. In this case, it is optial to reject a class i custoer and accept a class j custoer in state x at stage n, whereas the reverse is true at stage k. It follows that the ordering of the booking liits b in in i ay not be the sae for all n. In this case, the nesting of the classes ay be tie dependent. As we shall see in the next section, this phenoenon can occur when the fare in each class is adjusted by subtracting the expected cost of cancellation and no-show refunds. REMARK 4. As noted above, to reduce the coputational burden, it ay be advantageous to introduce a axiu overbooking level v (called the overbooking pad), resulting in an additional state constraint, 0 x C v, at each stage n. The qualitative properties of optial policies, such as onotonicity of an optial policy and the existence of booking liits, continue to hold in this setting. To see that this is the case, note that we can incorporate this constraint into the recursions as follows. At stage n 1, given the value functions U n 1 (x), for 0 x C v, define U n 1 (x) : U n 1 (C v) M(x C v), for x C v, where M is a positive nuber such that M ax i,n {r in }. Note that this definition ensures that booking requests will always be rejected in state C v at stage n, while preserving the properties of onotonicity and concavity of U n 1 and hence the onotonicity properties of an optial policy at stage n. It then follows by the inductive arguent given previously that the function U n (x) will also be nonincreasing and concave for 0 x C v. Now, define the opportunity cost functions u n 1 (x) : U n 1 (x) U n 1 (x 1), 0 x C v. Because an optial policy now rejects all booking requests in state C v, the optiality Eqs. 2 and 3 reduce to U n x p in ax r in u n 1 x, 0 xq n U n 1 x 1 1 xq n U n 1 x, 0 x C v 1, n 1, (8) U n C v C v q n U n 1 C v 1 1 C v q n U n 1 C v, n 1, (9) U 0 x E Y x, 0 x C v. (10) We shall use the optiality equations in this for to solve for the optial policy in the nuerical exaples in Section GENERAL MODEL WITH CLASS-DEPENDENT CANCELLATION AND NO-SHOW RATES (MODEL 2) IN THIS SECTION, we introduce our ost general odel (Model 2), which extends Model 1 by allowing class-dependent cancellation and no-show probabilities, as well as refunds at the tie of cancellation or no-show. As with Model 1, there are fare classes and N stages, nubered in reverse order, n N, N 1,..., 1, 0. Let x i denote the nuber of seats currently reserved by class i custoers, i 1,...,. Our state variable is now x (x 1,...,x ), the reservation vector. Let p in (x), q in (x), and p 0n (x), respectively, denote the probabilities of a request for a seat in fare class i, a cancellation by a custoer of class i, and a null event in period n, given the reservation vector x. Once again, we assue that one and only one of these events occurs during each stage. Note that we are now allowing the arrival probability to depend on the current syste state, as well as distinguishing between cancellations in different fare classes and allowing the probability of a cancellation in class i to depend upon the detailed state description, x (x 1,...,x ), rather than just the total nuber of seats booked, x i x i. (In applications, as we shall see, the class i cancellation probability will typically depend on x through x i, the nuber of seats currently reserved in class i.) By our assuption that, at ost, one request or cancellation can occur at each stage, we have p in x q in x p 0n x 1, for all x and n 1. If the event occurring at stage n is the arrival of a seat request, the syste controller (the booking agent) decides whether or not to accept the request, based on the fare-class i of the custoer and the current state x. If the event is a cancellation or null event, then no decision is ade. A custoer whose request for a seat in fare class i is accepted at stage n pays a fare rˆ in. A custoer in class i who cancels in decision period n receives a refund c in. With regard to no-shows, we now assue that, at the tie of departure, each custoer of class i has a probability i of being a no-show, dependent on the class i but independent of everything else. A class i

7 OVERBOOKING, CANCELLATIONS, AND NO-SHOWS / 153 custoer who is a no-show is refunded an aount d i. Let Y i (x i ) denote the nuber of people of class i who show up for the flight, given that the nuber of reserved class i seats is x i just before departure, so that x i Y i (x i ) is the nuber of no-shows in class i. Because each class i custoer has a probability 1 i of showing up for the flight, it is clear that Y i (x i ) has a binoial-(x i,1 i ) distribution. Let Y(x) : Y i (x i ) denote the total nuber of custoers who show up for the flight. If, at the tie of departure, Y(x) y, we incur an overbooking penalty (y). As before, we assue that (y) is non-negative, convex, and nondecreasing in y 0, with (y) 0 for y C. Our objective is to axiize the expected total net benefit of operating the syste over the horizon fro period N to period 0, starting fro state x (0,...,0),that is, with no seats booked, at the beginning of period N. The proble can be forulated as an MDP with stages n N, N 1,..., 0, in which the state, x (x 1,...,x ), is the current reservation vector. Once again, because our odel allows for the possibility of overbooking, cancellations, and no-shows, the state variable need not satisfy the constraint x i x i C. However, because we start with no seats booked at stage N and, at ost, one seat request can be accepted at each stage (because, at ost, one arrives), it follows that x i x i N n at stage n. Let X n : {x (x 1,...,x ): x i 0, i 1,...,; i x i N n}. Thus, at each stage n, there is an iplicit constraint, x X n. (As with Model 1, one can also introduce the constraint, i x i C v, where v is an overbooking pad.) As a function of the state, x X n, in period n, let Û n (x) denote the axial expected net benefit of operating the syste over periods n to 0. The optial value functions, Û n, are deterined recursively by Û n x p in x ax rˆ in Û n 1 x e i, Û n 1 x q in x c in Û n 1 x e i p 0n x Û n 1 x, x X n, n 1; (11) E i Û 0 x Y x x i Y i x i d, x X 0, (12) where e i is the ith unit -vector. The structure of this MDP reveals that the seatallocation proble is again essentially equivalent to a proble of adission control to a queueing syste, in which the custoers are the requests for seats and the cancellation process plays the role of service echanis. Now, however, because the different classes of custoers have different cancellation (service) rates and refunds, it is necessary to keep track of the nuber of custoers in each class, rather than just the total nuber, as a cancellation fro a reservation vector heavily laden with passengers prone to cancel has a higher probability of occurrence than one fro a plane filled with those unlikely to do so. We now show how to transfor the optiality Eqs. 11 into an equivalent set of equations in which all expected costs (caused by cancellations and noshows) are assessed at the instant of adission (booking of a seat) along with the reward (payent of the fare). To do this, we borrow a technique (the equivalent charging schee) fro the queueing-control literature, first proposed in Lippan and Stidha (1977). This transforation will facilitate the reduction of the proble to a one-diensional MDP in certain cases. Let H n (x) denote the total expected loss of revenue over periods n to 0 caused by cancellations and no-shows. (Another interpretation for H n (x) is that it is the negative of the value function for the policy that rejects all arrivals, starting fro state x at stage n.) Then, the functions H n are given by the recursive equations H n x p in x H n 1 x i 0 H 0 x E q in x c in H n 1 x e i, i x i Y i x i d x X n, n 1 (13) i x i d i, x X 0. (14) Now, define U n (x) : Û n (x) H n (x), n 0. Then, U n represents the axial expected controllable net benefit of operating the syste over periods n to 0, because H n (x) is an unavoidable loss of revenue. Using Eqs. 11, 12, 13, and 14, we obtain the follow-

8 154 / J. SUBRAMANIAN, S. STIDHAM, JR., AND LAUTENBACHER ing recursive optiality equations satisfied by the functions U n. U n x p in x ax rˆin H n 1 x e i H n 1 x U n 1 x e i, U n 1 x q in x U n 1 x e i p 0n x U n 1 x, x X n, n 1; (15) U 0 x E Y x, x X 0. (16) These equations are equivalent to the original optiality Eqs. 11 and 12 in the sense that they generate the sae optial booking policy. It follows fro Eq. 15 that this policy takes the for accept a class i request in stage n with reservation vector x N rˆ in H n 1 x e i H n 1 x U n 1 x U n 1 x e i. That is, in deterining whether or not to accept a seat request in fare-class i, we should first calculate the expected net revenue fro the booking by subtracting the arginal expected cancellation cost, H n 1 (x e i ) H n 1 (x), fro the gross fare received, rˆ in. This quantity should then be copared to the opportunity cost of the booking, U n 1 (x) U n 1 (x e i ), and the request should be accepted if and only if the forer exceeds the latter. In principle, the recursive optiality Eqs. 15 and 16 can be used to calculate the optial value functions, U n, and the associated optial booking policy, for each fare class i, reservation vector x X n, and stage n 0, 1,..., N. (This is just the standard backwards-recursive algorith of dynaic prograing.) However, for our odel, in its present, very general for, this algorith will often not be coputationally feasible because of the curse of diensionality specifically, the cobinatorial explosion in the nuber of possible states, x (x 1,..., x ), which grows exponentially in and N. (However, it is feasible to solve sall probles exactly: see Exaple 5 in Section 3.) For this reason, we look for siplifying but realistic assuptions that will reduce the size of the state space. To this end, we begin by aking the following assuption. ASSUMPTION 1. q in (x) q in (x i ), for all x (x 1,..., x ), where q in (0) 0 and q in (x i ) is nondecreasing in x i 1, i 1,...,, n N, N 1,..., 1. Assuption 1 says that the probability of a class i cancellation in a particular period depends only on the nuber of class i seats currently booked, and not on the reservations in other classes. This is a reasonable assuption in practice. The following lea can be proved easily by induction on n 1,..., N. LEMMA 2. Under Assuption 1, H n x H in x i, n 0, (17) where the functions H in satisfy the recursive equations (i 1,..., ) H in x i 1 q in x i H i,n 1 x i q in x i c in H i,n 1 x i 1, x i 0, n 1 (18) H i0 x i i x i d i, x i 0. (19) Now, let G in (x i ): H i,n 1 (x i 1) H i,n 1 (x i ). It follows fro Eq. 17 that G in (x i ) H n 1 (x e i ) H n 1 (x), the arginal expected cancellation cost associated with a fare-class i booking in state x at stage n, which appears in Eq. 15 and was discussed above. Moreover, it follows fro Eqs. 18 and 19 that the functions G in satisfy the recursive equations (i 1,..., ) G in x i q i,n 1 x i 1 q i,n 1 x i c i,n 1 1 q i,n 1 x i 1 G i,n 1 x i q i,n 1 x i G i,n 1 x i 1, x i 0, n 2 (20) G i1 x i i d i, x i 0. (21) Further siplifications occur if we strengthen Assuption 1, by requiring that the cancellation rate in each fare class in each period be proportional to the nuber of seats currently booked in that class: ASSUMPTION 1. q in (x) x i q in, for all x, where q in 0, i 1,...,, n N, N 1,...,1. As we shall see (Appendix A), this assuption is reasonable in applications of our discrete-tie odel to a syste operating in continuous tie, in which booking requests arrive according to tiedependent Poisson processes (cf. Lee and Hersh,

9 OVERBOOKING, CANCELLATIONS, AND NO-SHOWS / ). In that context, the assuption is equivalent to assuing that each custoer cancels independently of all other custoers, with a cancellation rate solely dependent on the custoer s class. Using induction on n in Eqs. 20 and 21, we iediately obtain the following lea. LEMMA 3. Under Assuption 1, G in (x i ) G n (i), independent of x i, where the functions G n (i) satisfy the recursive equations (i 1,..., ) G n i q i,n 1 c i,n 1 1 q i,n 1 G n 1 i, n 2; (22) G 1 i i d i. (23) This result akes sense intuitively. It states that the arginal expected cancellation cost associated with accepting a request for a seat in fare-class i in stage n is siply the expected cancellation cost attributable to that particular custoer over the reaining horizon, which is independent of the nuber of seats already booked. (It is not difficult to solve Eqs. 22 and 23 for an explicit expression for G i, but the expression is coplicated and we shall not need it. The recursive Eqs. 22 and 23 are ore suitable for nuerical calculations.) To recapitulate, we have shown how the calculation of the arginal expected cancellation cost, H n (x e i ) H n (x), siplifies under Assuption 1, a realistic assuption about cancellation rates. In this case, an optial seat-allocation policy can be found by choosing, in state x at stage n, the axiizing action in the recursive optiality equations, which now take the for U n x p in x ax rˆin G n i U n 1 x e i, U n 1 x x i q in U n 1 x e i p 0n x U n 1 x, x X n, n 1 (24) U 0 x E Y x, x X 0. (25) An optial policy now takes the for accept a class i request in stage n with reservation vector x N rˆ in G n i U n 1 x U n 1 x e i. REMARK 5. Using terinology fro queueing control (borrowed in turn fro welfare econoics), the arginal expected cancellation cost, G n (i), is an internal effect of the acceptance of the custoer (booking request). It is a cost associated directly with the accepted custoer, naely, the expected aount that will be refunded to that custoer resulting fro cancellation or no-show. By contrast, the opportunity cost, U n 1 (x) U n 1 (x e i ), is an external effect, in that it represents the expected loss of revenue fro future custoers, caused by the adission of the custoer in question. So, we see that an optial policy adits the custoer (accepts the booking request) if and only if the gross fare exceeds the su of the internal and external effects. We are still left with the necessity of evaluating the optiality Eqs. 24 and 25 for each state x (x 1,...,x ) X n, for each stage n. In other words, the curse of diensionality is still with us. Our next result shows that we can reduce the proble to an equivalent proble with a one-diensional state variable (the total nuber of booked seats), provided that arrivals of booking requests are independent of the nuber of seats already booked, and the individual cancellation rates and no-show probabilities are independent of the fare class. ASSUMPTION 1. q in (x) x i q n, for all x X n, i 1,...,, where q n 0, n N, N 1,..., 1. ASSUMPTION 2. p in (x) p in, for all x X n, i 1,...,, n N, N 1,..., 1. ASSUMPTION 3. i, i 1,...,. Let r in : rˆ in G n (i). THEOREM 1. Under Assuptions 1, 2, and 3, the optial value functions, U n (x), depend on x only through x i x i, and are deterined by the recursive optiality equations U n x p in ax r in U n 1 x 1, U n 1 x xq n U n 1 x 1 1 p in xq n U n 1 x, 0 x N n, n 1 (26) U 0 x E Y x, 0 x N, (27) where Y(x) Bin(x, 1 ). Proof. The proof is by induction on n. By assuption, U 0 (x) U 0 (x) E[ (Y(x))] and so depends on x only through x i x i. Let n 1 and suppose

10 156 / J. SUBRAMANIAN, S. STIDHAM, JR., AND LAUTENBACHER U n 1 (x) U n 1 (x) for all x. Then, it follows fro Eq. 26 and Assuptions 1, 2, and 3 that U n x p in ax r in U n 1 x 1, U n 1 x x i q n U n 1 x 1 1 p in xq n U n 1 x, n 1, (28) where we have used the fact i p in i x i q n p 0n (x) 1. It follows fro Eq. 28 that U n (x) U n (x) and Eq. 26 holds. This copletes the induction and the proof of the theore. e The proble represented by the optiality Eqs. 26 and 27 now has exactly the for of the onediensional proble (Model 1) discussed in Section 1, with r in rˆ in G n (i), and q n (x) xq n (a concave function of x). So, all the onotonicity results for Model 1 apply, in particular, the optiality of a booking-liit policy. REMARK 6. Note that we were able to reforulate the proble with the one-diensional state variable x even though the cancellation and no-show costs are still allowed to be class dependent. This would not have been possible had we continued to charge cancellation and no-show costs in the periods in which they occur, rather than charging the expected cancellation and no-show cost during the period in which the seat is booked. REMARK 7. In ost YM applications, rˆ in r i. That is, the gross fare for class i is independent of n. Assue (without loss of generality) that the classes are ordered so that r 1 r 2 r. Without cancellations, it is well known that the optial booking liits are nested in the sae order as the classes, b 1n b 2n b n, and this nesting is the sae for all stages n. Aswe saw in Section 1, however, when the fares r in are tie dependent, the ordering of the booking liits ay be different for different stages. In particular, this ay happen in the present case, in which r in r i G n (i). Consider two fare classes, i and j, with r i r j (so that i j). If r i G n (i) U n (x) U n (x 1) r j G n (j) (which can occur if G n (i) G n (j) r i r j ), then it will be optial to reject fare class i and siultaneously accept fare class j in period n, even though class i has the higher (gross) fare. This could happen if c ik c jk, for all n k 1, (e.g., if fare class i is fully refundable and fare class j is nonrefundable). REMARK 8. We recognize that it is not realistic to assue that cancellation and no-show rates do not depend on class. We believe, however, that our odel is a significant iproveent over previous odels, in that it explicitly odels the effects of cancellations on future seat availability, as well as class-dependent refunds. The class-dependence of refunds perits accurate odeling of the internal effect, G n (i), of accepting a request (cf. Reark 5 above), whereas the dependence of cancellation and no-show rates on class influences the solution only through the calculation of the external effect, U n (x) U n (x 1). Typically, the internal effect is a first-order effect, whereas the external effect is a second-order effect. Therefore, we have reason to hope that assuing class-independent cancellation and no-show rates (as we do, for exaple, with our proposed heuristic approxiation in Appendix B) will still give us a good approxiation to the optial solution, while retaining the coputational advantage of a one-diensional state variable. Our nuerical results (see Exaple 6 in the next section) tend to support this hope, while ephasizing the need for care in choosing the paraeters in the heuristic approxiation. 3. NUMERICAL EXAMPLES IN THIS SECTION, we report the results of nuerical solution of several exaples. We divide the results into four subsections, each highlighting a different aspect of the work in this paper. Section 3.1 provides counterexaples to coonly held notions in the YM literature, such as the onotonicity of the booking liits and bid prices in the nuber of booking periods reaining. In Section 3.2, we provide an exaple of our algorith perfored on actual airline data. We deonstrate that the MDP ethod can handle real-world-size probles. Section 3.3 includes a odel of odest size in which the cancellation rates are class-dependent. This deonstrates the coputational feasibility of the ultidiensional MDP odel of Section 2 for sall probles. Finally, in Section 3.4, we deonstrate the power of odeling cancellations through an exaple that copares various approaches to solving YM probles in the presence of class-dependent cancellation

11 OVERBOOKING, CANCELLATIONS, AND NO-SHOWS / 157 and no-show rates. We show that significant increases in revenue can result fro fully taking into account the effects of cancellations and no-shows. For all the exaples, we work with the special case in which r in r i G n (i), where r i is the fare in class i (invariant with n) and G n (i) is the expected loss of revenue resulting fro cancellation or no-show (defined recursively by Eqs. 22 and 23) in other words, the version that results fro applying the equivalent-charging transforation to Model 2 under Assuption 1. For coputational purposes, we introduce an overbooking pad v. When cancellation and no-show rates are independent of the class (Assuption 1 ), the result is a version of Model 1 (as we saw in Section 2). The recursive optiality equations in this case take the for U n x p in ax r i G n i u n 1 x, 0 xq n U n 1 x 1 1 xq n U n 1 x, 0 x C v 1, n 1, U n C v C v q n U n 1 C v 1 1 C v q n U n 1 C v, U 0 x E Y x, 0 x C v. n 1, (cf. Reark 4). These are the equations that we use for the nuerical solution of the one-diensional exaples (Sections 3.1 and 3.2). For the ultidiensional exaples (Sections 3.3 and 3.4) we use the corresponding special case of Eqs. 24 and 25, with the addition of an overbooking pad v. Paraeters TABLE I Paraeters for Exaple 1 Period n p 1n p 2n q n ) for the plots corresponding to x 7 and x 12, and that the forer plot is always above the latter. The capacity reaining is also identical (s 2) for the plots corresponding to x 3 and x 8, and the forer plot is always above the latter. This indicates two things: (1) when we have the possibility of cancellations, the functions u n and the optial policy depend on the total capacity, C, and the capacity reaining, s C x; and (2) for a given s, the u n functions are onotone (nonincreasing) in C. It is clear fro Figure 2 that the booking liits for class 2 are not onotonic in n. EXAMPLE 2. This exaple is taken fro Lee and Hersh (1993). We consider four booking classes with fares r 1 200, r 2 150, r 3 120, and r The capacity of the airplane is C 10. There are no cancellations, no-shows, and no overbooking. The values of the paraeters p in, i 1, 2, are shown in Table II. (Because Lee and Hersh do not allow cancellations, q n 0, for all n.) Figure 3 plots u n (x) versus n for different x values. In this exaple, the u n functions are onotone in n because of the absence of cancellations. 3.1 Counterintuitive Results EXAMPLE 1. Exaple 1 shows that the functions u n (x) behave counterintuitively in that they are not always onotone in n. We consider two booking classes with fares r and r The refund aounts are c in r i, i 1, 2. The axiu overbooking level is v 3. The overbooking cost is given as 55 per seat overbooked. There are no noshows. The values of the paraeters p in, i 1, 2, and q n are shown in Table I. Figure 1 plots the bid prices, u n (x) with respect to n for two different capacities, C 5 and C 10, and various fixed values of x. Note first that the functions u n (x) are not onotonically increasing in n, in contrast to the exaples in Lee and Hersh (1993). Second, if we observe carefully, we note that the capacity reaining, s C x, is identical (s Fig. 1. u n (x) versus n for different x values for Exaple 1.

12 158 / J. SUBRAMANIAN, S. STIDHAM, JR., AND LAUTENBACHER Paraeters TABLE III Paraeters for Exaple 3 Period n p 1n p 2n p 3n p 4n q n Fig. 2. Partition of the state space by critical values for Exaple 1. Paraeters TABLE II Paraeters for Exaple 2 Period n p 1n p 2n p 3n p 4n EXAMPLE 3. We consider four booking classes with fares r 1 200, r 2 150, r and r The capacity of the airplane is C 20. The overbooking cost is a piecewise linear and concave function of the overbooking level. The axiu overbooking level is v 10. Cancellations take place at rate q n. On cancellation, each fare class is refunded a different aount c in, c 1n 200, c 2n 120, c 3n 60, and c 4n 0. Class 4 is non-refundable. There are no no-shows. The values of the paraeters p in, i 1, 2, and q n are shown in Table III. Figure 4 shows a plot of u n (x) versus n for different x values, whereas Figure 5 plots u n (x) versus x for different values of n. Note that u n (x) is nondecreasing in x. Figure 6 partitions the state space into the various acceptance regions. 3.2 Real Airline Data EXAMPLE 4. (This exaple was previously reported in SUBRAMANIAN, CAMPBELL, and PHILLIPS (1996).) We obtained airline deand and capacity data fro a ajor U.S. airline. To protect the interests of this airline and to guard against release of sensitive data, we have held back (or odified) certain details. The following exaple uses this data to construct a realistic airline YM proble for a single leg with an airplane of capacity C 100, six fare classes, and a 331-day booking horizon. Fig. 3. u n (x) versus n for different x values for Exaple 2. Fig. 4. u n (x) versus n for different x values for Exaple 3.

13 OVERBOOKING, CANCELLATIONS, AND NO-SHOWS / 159 Fig. 7. Booking liits versus tie for Exaple 4: real airline data. Fig. 5. u n (x) versus x for different n values for Exaple Calculation of Paraeters We received historical airline deand data for all traffic on one flight leg for several departure dates (including both local and ultileg traffic). The deand data are given by booking-class, snapshot, point of sale, and passenger type (i.e., individual or group). For a given date, we selected only the local traffic for individual bookings, and restricted point of sale to the United States. We calculated the arrival and cancellation paraeters as follows. 1. Each snapshot was divided into subperiods. 2. The booking probability for fare class i in subperiod n (in snapshot k) was calculated as p in nuber of bkgs in fare class i and kth snapshot N k. 3. The cancellation rate in subperiod n was calculated as q n total cancellations in kth snapshot. N k load booked at the beginning of kth snapshot The nuber of subperiods N k was calculated so that p in C v q n 1. Fig. 6. Partition of the state space by critical values for Exaple Results Figure 7 shows the plot of the booking liits versus the tie for each of the booking classes. To understand clearly the dynaics of the bid price with respect to tie, it will be helpful to refer to Table IV, which shows a apping of the booking periods to weeks-before-departure for this exaple. Note that the booking periods grow shorter as departure approaches. This is natural because the event density (expected nuber of events per unit tie) typically increases as takeoff nears. In Fig. 7, the six booking classes are lettered in decreasing order of fare, with class A representing the highest fare class and class F the lowest.

14 160 / J. SUBRAMANIAN, S. STIDHAM, JR., AND LAUTENBACHER TABLE IV Mapping of Periods to Weeks before Departure Snapshot wbd period wbd, weeks before departure Coputation-Tie Analysis Solving a single-leg proble with 100 seats and 500 periods on a PC with a 100-MHz Intel 486 processor takes approxiately ten seconds. This tie increases roughly as the product of the nuber of seats and nuber of periods in the odel. We are currently investigating perforance on a high-end UNIX server and are pursuing other strategies to iprove run tie. The desired target is to solve a single-leg proble of this agnitude in one second. This would translate to recalculation of 500 legs for 100 departure dates on a 6-processor UNIX server in roughly one hour. 3.3 Class-Dependent Cancellations EXAMPLE 5. We now solve an exaple in which the cancellation rates are class dependent. Although the size of the odel is odest, this does not entirely diinish the usefulness of the results. In larger probles, it is typically the end effects when both the tie reaining until departure and the reaining capacity are sall that are ost crucial to the perforance of a booking policy. We consider three booking classes, with fares r 1 5, r 2 3, and r 3 1. The airplane capacity is 30, with an overbooking pad of 2. The booking horizon is 40 periods. Class 1 is fully refundable, c 1n d 1 5, whereas class 2 receives only a partial refund, c 2n d 2 1, and class 3 is nonrefundable, c 3 d 3 0. The penalties for overbooking by one and two seats, respectively, are 4 and 10. The values of the paraeters p in and q in, i 1, 2, 3 are given in Table V. In this exaple, class 1 is intended to be a fullyrefundable full-fare ticket, class 2 a ildly discounted ticket with soe cancellation restrictions, Paraeters TABLE V Paraeters for Exaple 5 Period n p 1n p 2n p 3n q 1n q 2n q 3n and class 3 a discount fare with both purchase and cancellation restrictions. Finding the optial booking policy for this exaple took approxiately 30 seconds of CPU tie on a achine with a 200-MHz Pentiu processor running the Linux operating syste. The details of the optial value functions, bid prices, and optial booking liits are available fro the authors upon request. We have oitted the here because of the difficulty in displaying the output fro a ultidiensional odel in a copact yet eaningful way. (In Exaple 6 below, we will exaine the optial value functions and optial booking liits for a saller ultidiensional exaple.) Here we do show, however, that class-dependent cancellation rates can cause the nesting of the net fare to be tie dependent (cf. Rearks 3 and 7 above). Table VI shows the net fare of the three fare classes for the final periods in the booking horizon. Note that the nesting order of classes 1 and 2 reverses toward the end of the horizon. 3.4 Power of Cancellations EXAMPLE 6. The cornerstone of the approach deonstrated in this paper is the inclusion of custoer cancellations. To highlight the effects of allowing cancellations, we consider a sall exaple with class-dependent cancellation and no-show rates. Available capacity is C 4 with an overbooking pad of v 2. There are 2 classes, with r 1 3 and r 2 1. Class 1 is fully refundable, c 1n d 1 3, whereas class 2 is nonrefundable, c 2n d 2 0. Penalties of 2 and 6 correspond to overbooking levels of 1 and 2, respectively. The reaining paraeters are suarized in Table VII. We copare four ethods for solving this exaple. 1. Copletely ignore cancellations. TABLE VI Net Fare versus n for Exaple 5 Period n Class

15 OVERBOOKING, CANCELLATIONS, AND NO-SHOWS / 161 Paraeters TABLE VII Paraeters for Exaple 6 Period n p 1n p 2n q 1n q 2n Method TABLE VIII Coparison of Class-Independent Cancellation Rates for Exaple 6 Period n a b c Adjust fares by expected lost revenue due to cancellations, but ignore effects of cancellations on future seat availability. 3. Approxiate the class-dependent cancellation and no-show rates with a single class-independent rate. 4. Solve the proble as stated. Method 1 is the easiest ethod and reflects the general approach in the YM literature. The effects of cancellations and overbooking on both the net fare received and on the nuber of passengers booked are ignored. [Following the previous literature, however, we do add a pad to the physical capacity and then allocate this liit without cancellations or overbooking as if it were the true capacity. In effect, this approach decoposes the proble, first setting an overbooking liit (the pad), perhaps based on a siplified cancellation and overbooking odel, then applying a seat allocation odel that assues no cancellations or overbooking.] Method 2 is a first step toward incorporating cancellations explicitly. Fares are adjusted by subtracting the (readily calculated) expected lost revenue. The effects of cancellations on the state variable are still oitted fro the MDP optiality equations. Method 3 incorporates both the reduction in net fare and the fluctuations of total passengers booked produced by cancellations, but does so by introducing a single cancellation and no-show rate that is assued to hold across all classes. Note that Methods 1 3 use one-diensional MDP odels (Model 1). Method 4 solves the proble optially as stated, using the ultidiensional MDP odel (Model 2). When using Method 3, for the purposes of coparison, we solved the proble with three different rates, suarized in Table VIII. For Method 3a, we averaged the cancellation and no-show rates of the two classes. In Method 3b, we siply used the rates corresponding to class 1. We chose the class-independent rates in Method 3c in such a way as to approxiate as closely as possible the solution obtained in Method 4. In this particular exaple, that consisted of using 40% of the class-1 cancellation and no-show rates. We copare the four ethods by the following procedure. For Methods 1 3, we solve the siplified, one-diensional proble optially. The optial booking liits fro each of these ethods are then evaluated in the context of the actual (ultidiensional) proble, and the results copared to those of the optial solution as obtained through Method 4. The expected revenues obtained by each of the ethods are suarized in Table IX, where % Sacrificed is the additional revenue that could be gained by solving the proble optially (Method 4), expressed as a percentage of the revenue fro the given ethod. Note that Method 2 actually exhibits worse perforance than does Method 1, even though it appears to be a ore accurate odel in that cancellation and no-show refunds are subtracted fro the gross fare. Evidently, adjusting the fare in this way without also taking into account the effects of cancellations on future seat availabilities sends the wrong signal to the algorith for this exaple. To understand why this ight be the case, consider a class i 1 with a large cancellation/no-show refund and a high cancellation rate. After subtracting the expected cancellation and no-show refunds fro the fare, the net fare for class i ay be so low that it ends up being rejected to save seats for later arriving higher-fare requests. But, because Model 2 does not take into account the fact that a class i custoer, if accepted, has a high probability of canceling later (thus aking the seat available anyway), it tends to penalize this class ore than it should. This result indicates the iportance of accurately odeling both the internal and the external effects of cancellations, as we do in Models 3c and 4, for TABLE IX Suary of Methods 1 4 Method U 16 (0, 0) % Sacrificed a b c

16 162 / J. SUBRAMANIAN, S. STIDHAM, JR., AND LAUTENBACHER Fig. 9. Class-2 booking liits for the different ethods for Exaple 6. Fig. 8. U n (0) versus n for the different ethods for Exaple 6. exaple. In particular, the YM practitioner should be wary of siplistic approaches, such as siply calculating the net fare by subtracting the expected cancellation and no-show refund fro the gross fare and then using one of the standard YM approaches, such as EMSR or Lee and Hersh s odel. Such an approach, as we have seen in this exaple, could actually lead to lower revenue. Figure 8 copares the values of U n (0) for each of the four ethods. Note how iportant it is to choose the single-leg cancellation rate properly. (Periods 0 5 are oitted fro the graph because all 4 ethods produce identical values of U n (0, 0) during those periods.) Figure 9 plots the optial booking liits for class 2 for Methods 1 4. In this figure, we use booking liit to ean the axiu nuber of reservations to accept in a given period. Thus, a booking liit of 4 iplies that a passenger would be accepted provided 3 or fewer passengers had been accepted previously. Because Method 4 uses class-dependent cancellation and no-show rates, the optial booking liits depend on the nuber of passengers booked in each class. In Figure 9, we have rounded down the policy for Method 4. For exaple, if we put down the booking liit as three, that eans that we always accept whenever there are two or fewer custoers in the syste, regardless of how they are distributed. It is possible that we ight also accept requests when there are ore than two custoers, but only if they are distributed in a certain way. Table X displays the optial booking liits for class 2 custoers. Note the slight deviations fro Figure CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK WE HAVE CONSIDERED a single-leg airline YM proble with ultiple fare classes and tie-dependent arrival probabilities. Currently booked custoers ay cancel at any tie or becoe no-shows at the tie of departure, with probabilities and refunds that, in general, ay be both class and tie dependent. Overbooking is peritted. We have forulated the proble as a discretetie MDP and used dynaic prograing (backward induction) to analyze it. The analysis has two coponents: (1) characterization of the structure of an optial policy; and (2) nuerical calculation of the paraeters of an optial policy. Our analysis exploits the equivalence of the YM proble to the well-studied proble of optial control of adission to a queueing syste. In its ost general for (Model 2), the MDP has a ultidiensional state space, because it is necessary to keep track of the nuber of seats booked in each fare class. When cancellation and no-show probabilities are independent of the fare class, we have used a transforation (the equivalentcharging schee) to convert the proble to an equivalent proble (Model 1) with a one-diensional state variable: the total nuber of seats booked. In this case, we have used well-known results fro queueing-con-

17 OVERBOOKING, CANCELLATIONS, AND NO-SHOWS / 163 TABLE X Booking Liits for Class 2 fro Method 4 Period n (x 1, x 2 ) (0, 0) (1, 0) (0, 1) (2, 0) (1, 1) (0, 2) (3, 0) (2, 1) (1, 2) (0, 3) (4, 0) (3, 1) (2, 2) (1, 3) (0, 4) (5, 0) (4, 1) (3, 2) (2, 3) (1, 4) (0, 5) (6, 0) (5, 1) (4, 2) (3, 3) (2, 4) (1, 5) (0, 6) , accept; 0, reject. trol theory to show that an optial policy is onotonic in the state variable and thus is characterized by booking levels for each fare class. These booking levels are deterined by the optial bid prices, which are easily characterized in ters of the dynaic-prograing optial value functions. In our nuerical exaples, we have deonstrated that an optial booking policy can have counterintuitive properties when cancellations and no-shows are included. For exaple, booking liits need not be onotonic in the nuber of booking periods reaining. Also, the opportunity cost of a booking now ay depend on both the reaining capacity and the nuber of seats already booked, rather than just the reaining capacity. Using data fro a real airline application, we have shown that our one-diensional odel is coputationally feasible for realistic-size probles, in ters of both the availability of data and the running ties of the algorith. We have also deonstrated coputational feasibility for the ultidiensional odel on odest-size probles, which suggests that it ight be used to explore the end effects as both the reaining horizon length and the reaining capacity becoe sall. Our experience and that of other researchers suggest that this is where the action is, in the sense that revenues are ost sensitive to odel accuracy in this region. Finally, we have considered a sall proble with class-dependent cancellation and no-show rates and copared the perforance of four ethods. In order of increasing accuracy (and coplexity), these are: (1) Model 1, ignoring cancellations and no-shows (essentially the odel of Lee and Hersh, 1993); (2) Model 1, using equivalent charging subtracting the expected cancellation and no-show refunds fro the gross fare but ignoring the (probabilistic) effects of cancellations on future seat availabilities; (3) Model 1, incorporating both cancellation and noshow refunds and probabilities, but using approxiate probabilities independent of class; (4) Model 2, with both refunds and probabilities dependent on class (as given). For this exaple, revenue actually decreases when going fro Method 1 to Method 2. An increase of nearly 9% can be achieved by incorporating both refunds and cancellation and no-show

18 164 / J. SUBRAMANIAN, S. STIDHAM, JR., AND LAUTENBACHER probabilities, with careful use of the one-diensional heuristic approxiation (Method 3). The revenue increent in going fro Method 3 to Method 4 is less than 1%, which suggests that the full ultidiensional odel ay not always be needed. Additional nuerical work is needed to see if these qualitative conclusions are valid over a wider range of paraeter values. In principle, the ethodology of this paper can be applied to the ulti-leg YM proble. The hurdle is the cobinatorial explosion of the state space in the nuber of legs. We are currently working on solving the two-leg proble exactly and efficiently. For the general ulti-leg proble, we hope to develop efficient heuristics to approxiate the optial policy. APPENDIX A: APPLICATION TO SYSTEM WITH TIME-DEPENDENT POISSON ARRIVAL PROCESS IN THIS APPENDIX, we show how to apply the MDP odel of Section 1 to a continuous-tie seat-allocation proble. In the continuous-tie proble, the decision horizon is the tie interval [0, T]. Seat requests for fare class i (i 1,..., ) arrive according to a tie-dependent Poisson process with rate i (t), 0 t T. So, the probability that a class i request occurs in the interval (t, t dt) is i (t)dt o(dt), independent of the previous arrivals and everything else that occurred in [0, t). Cancellations occur in a eoryless, possibly tie-dependent, anner. Specifically, each custoer in the syste at tie t has a probability (t)dt o(dt) of canceling in (t, t dt), independent of everything that occurred in [0, t). A custoer whose request for a seat in fare class i is accepted pays the fare r i (assued independent of t, to keep the exposition siple). A custoer in class i who cancels at tie t receives a refund c i (t), 0 t T. Each custoer who holds a seat reservation in class i just before departure has a probability i of being a no-show, in which case the custoer receives a refund d i,asin the discrete-tie odel. To approxiate this odel by a discrete-tie odel, we divide the tie interval [0, T] into N periods, where N is a sufficiently large integer. There are two ways of doing this, both of which lead to an MDP odel of the for presented in Section 2. The first ethod is an extension to tie-dependent processes of the approach introduced into queueing control by Lippan (1975) and coonly called uniforization. This approach results in periods of rando length, with an exponential distribution that is independent of the state but ay be tie dependent. The second approach is siply to divide the interval [0, T] into N deterinistic subintervals, possibly of varying lengths, each sall enough so that the probability of ore than one event (arrival or cancellation) in a subinterval is negligible. Approxiation using Uniforization In the uniforization approach the continuoustie syste is observed at discrete, randoly spaced points in tie, each point corresponding to the arrival of a seat request, a cancellation, or a null event. Null events, which do not change the state of the syste, are introduced fro a Poisson process with a state- and tie-dependent rate in such a way that the total rate at which the observation points occur is independent of the state. In addition, this total event rate and the total nuber of periods N are chosen large enough so that the expected horizon length is close to the actual horizon length T (by the law of large nubers). To construct the tie-dependent uniforization and the equivalent discrete-tie MDP, we ust specify the probabilities p in, i 1,...,, and q n, for each decision period n and state x. Beginning with period N, let in : i (0) (i 1,...,), N : in, and N : (0). Choose N N, and set p in in, i 1,...,, q N N. N N For each period n, N n 1, assue N, N 1,..., n 1 have been chosen and let t N N 1 n 1 Let c in : c i (t), in : i (t) (i 1,...,), n : in, and n : (t). Choose n n (N n) n, and set n n n, p in in n, i 1,...,, q n n. n Upon substitution of these expressions in Eqs. 4 and 5, the optiality equations for this proble take the for U n x 1 n n E V n x, R n x n U n 1 x 1 n n x n U n 1 x ], 0 x N n, n 1, (A1)

19 OVERBOOKING, CANCELLATIONS, AND NO-SHOWS / 165 V n x, r ax r U n 1 x U n 1 x 1, 0 U n 1 x, 0 x N n, n 1, (A2) U 0 x E Y x, 0 x N. (A3) Coparing Eqs. A1 and A2 to the optiality Eqs. 2 and 3 in Lippan and Stidha (1977), we see once again that our odel takes the sae for as the finite-horizon arrival-control odel in Lippan and Stidha (1977), with the arrival and service rates allowed to depend on the nuber of periods reaining. Specifically, with n periods reaining and x custoers in the syste, the tie Z n until the next event occurs is exponentially distributed with ean rate n. The next event is an arrival (booking request) with probability n / n, a departure (cancellation) with probability x n / n, and a null event with probability ( n n x n )/ n. (The requireent that n n (N n) n ensures that 0 ( n n x n )/ n 1. For the case of tie-hoogeneous paraeters, LIPPMAN (1976) has observed that the MDP that results fro uniforization can be used to solve approxiately a continuous-tie Markovian decision process with a finite horizon of fixed, deterinistic length T. More precisely, by choosing the uniforization paraeter and the nuber of periods, N, so that N/ T and then letting 3, N 3, one can show that optial policies for the discreteparaeter MDP converge to optial policies for the original continuous-tie MDP. The condition that N/ T ensures that the discrete-paraeter MDP has an expected horizon length T; oreover, as N and approach, the actual horizon length converges to T, by the law of large nubers. In addition, for any t, 0 t T, an approxiately optial policy to follow at tie t ay be found by calculating the optial policy for the discrete-paraeter MDP with n periods reaining, where (N n)/ t, when both N and are sufficiently large. Our continuous-tie seat-allocation proble differs fro that considered by Lippan (1976) in that it has tie-dependent paraeters in addition to a finite horizon. As a result, the discrete-paraeter MDP that we have constructed involves an additional level of approxiation. First, we approxiate the continuously varying paraeters, i (t) and (t), by paraeters in and n, that are constant over rando (exponentially distributed) intervals. Then we choose the nuber of periods, N, and the totalevent rate, n, for each period n N,..., 1 so large that the length of each such rando interval approaches zero. We expect that an approxiately optial policy to follow at tie t ay be found by calculating the optial policy for the discreteparaeter MDP with n periods reaining, where 1 1 N N N n 1 t, when N and N,..., 1, are sufficiently large. Although a theoretical proof that this approach provides approxiately optial policies for the continuous-tie proble with tie-dependent paraeters does not see to be available, we have proposed this approach as an intuitively plausible and coputationally feasible heuristic. In any case, in the scenario proposed by Lee and Hersh (1993), in which it is assued that the tie-dependent Poisson arrival processes for each fare class have constant arrival rates over fixed tie intervals (called booking periods), the validity of the approxiation follows by a straightforward extension of the arguents in Lippan (1976). Approxiation using Decision Periods of Deterinistic Length Here, we divide the horizon into N deterinistic intervals, which can, however, be of unequal length. The length of these intervals is chosen to be sall enough so that the probability of ore than one event (arrival/cancellation) occurring in an interval is sall, and the probability of each type of event can be approxiated by the ean rate for that event ties the length of the interval. Lee and Hersh (1993) follow essentially this approach, although they do not allow cancellations and no-shows. Suppose the length of the nth period is n, n N,..., 1, where, as usual, we nuber the periods in reverse chronological order. Beginning with period N, let in : i (0) (i 1,..., ), n : in, N : (0), and set p in in N, i 1,...,, q N N N. For each period n, N n 1, let t N N 1... n 1. Let in : i (t) (i 1,..., ), n : in, n : (t), and set p in in n, i 1,...,, q n n n. For sufficiently sall n, it follows fro the properties of the exponential distribution that these expressions for the discount factor n and the probabilities, p in and q n, are accurate within an error that is o( n ). Upon substitution of these expressions in Eqs. 4 and 5, the optiality equations for this proble take the for U n x n n E V n x, R n x n n U n 1 x 1 1 n x n n U n 1 x, 0 x N n, n 1, (A4)

20 166 / J. SUBRAMANIAN, S. STIDHAM, JR., AND LAUTENBACHER V n x, r ax r U n 1 x U n 1 x 1, 0 U n 1 x, 0 x N n, n 1, (A5) U 0 x E Y x, 0 x N. (A6) Note that these equations are in the sae for as the optiality Eqs. A1, A2, and A3 for the uniforization approxiation, as can be seen by replacing n by ( n ) 1 in Eq. A4. APPENDIX B: HEURISTIC MODEL FOR CLASS- DEPENDENT CANCELLATIONS AND NO-SHOWS RECALL THAT WE WERE able to transfor our MDP odel to an equivalent odel with a one-diensional state variable, provided that Assuption 1 holds. For convenience, we reproduce Assuption 1 below. ASSUMPTION 1. q in (x) x i q n, for all x, i 1,...,, where q n 0, n N, N 1,..., 1. This assuption ay not be realistic in applications. As an extree exaple, in the case of nonrefundable fares, the cancellation or no-show probability is close to zero, whereas cancellation and noshow rates tend to be positive and are often nonnegligible aong passengers who pay full fare. Assuption 1 (reproduced below) is ore realistic, inasuch as it allows cancellation rates to be class dependent. ASSUMPTION 1. q in (x) x i q in, for all x, where q in 0, i 1,...,, n N, N 1,..., 1. Assuption 1 ay be ore realistic, but then, we are faced with the proble of keeping track of an enlarged state x. To get around this proble, we propose a heuristic approxiation schee, in which we estiate the nuber of seats, x i, booked in each fare class i, based on the observed total nuber of seats x. Fro these estiates, we can copute q n (x), the estiated total cancellation rate, given that x seats are currently booked in period n. We suggest below two intuitive ways to estiate q n (x), given x and q in, i 1,...,. METHOD 1. Define q n x : f in xq in, (A7) where f in is the expected fraction of class i custoers with current reservations in period n. If we siplify the above equation, we obtain q n (x) x i f in q in xqˆ n, where qˆ n : i f in q in can be interpreted as the expected average cancellation rate per custoer in period n. METHOD 2. Define q n x : f in x q in, (A8) where f in (x) is the expected nuber of class-i reservations given that the total nuber of reservations is x. With the above setup, the optiality Eqs. 4, 5, and 6 hold, and the proble is seen to be equivalent to control of adission to a queueing syste. Note that, in Method 2, the service rate q n (x) is a possibly nonlinear function of x, the nuber of custoers present. As long as q n (x) is concave and nondecreasing in the state x, however, the inductive arguent (Theore 1 of Lippan and Stidha, 1977) continues to be applicable and iplies that the optial value function U n is concave and nonincreasing, so that the optial adission policy is onotonic in the state. When q n (x) i f in xq in qˆ nx (Method 1), then q n (x) is a linear function of x and, hence, trivially concave. But when we use Method 2 to estiate q n (x) i f in (x)q in, then we have to choose the functions f in (x) carefully so that q n (x) is concave. We plan to explore this approach in ore depth in a future paper. ACKNOWLEDGMENTS THE AUTHORS WOULD LIKE to acknowledge the contribution of Anatoly Shaykevich to an earlier version of this paper (Janakira, Stidha, and Shaykevich, 1994). Anatoly s econoic insights inspired us to develop the equivalent charging schee described in Section 2. We also thank R.L. Phillips (President and CEO, DFI.Aeronoics) and G.C. Capbell (Vice President, DFI.Aeronoics) for valuable assistance in developing the ideas in Section 3.2. We are grateful to G. Colville (currently at Delta Airlines) for giving us perission to use real airline deand and capacity data in our analysis. Finally, we thank Jeff Rutter (DFI.Aeronoics) for help with transferring and anipulating large volues of airline deand data. REFERENCES J. ALSTRUP, S. BOAS, O. MADSEN, AND R. VIDAL, Booking Policy for Flights with Two Types of Passengers, Eur. J. Oper. Res. 27, (1986). P. P. BELOBABA, Airline Yield Manageent: An Overview of Seat Inventory Control, Transp. Sci. 21, (1987).