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Revenue Management in a Dynamic Network EnvironmentDimitris Bertsimas, Ioana Popescu,

To cite this article:Dimitris Bertsimas, Ioana Popescu, (2003) Revenue Management in a Dynamic Network Environment. Transportation Science37(3):257-277. http://dx.doi.org/10.1287/trsc.37.3.257.16047

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Revenue Management in a DynamicNetwork Environment

Dimitris Bertsimas • Ioana PopescuSloan School of Management, MIT, Cambridge, Massachusetts 02139

INSEAD, Fontainebleau Cedex 77305, [email protected] • [email protected]

We investigate dynamic policies for allocating scarce inventory to stochastic demand formultiple fare classes, in a network environment so as to maximize total expected rev-

enues. Typical applications include sequential reservations for an airline network, hotel, orcar rental service. We propose and analyze a new algorithm based on approximate dynamicprogramming, both theoretically and computationally. This algorithm uses adaptive, nonad-ditive bid prices from a linear programming relaxation. We provide computational resultsthat give insight into the performance of the new algorithm and the widely used bid-pricecontrol, for several networks and demand scenarios. We extend the proposed algorithm tohandle cancellations and no-shows by incorporating oversales decisions in the underlyinglinear programming formulation. We report encouraging computational results that showthat the new algorithm leads to higher revenues and more robust performance than bid-pricecontrol.

IntroductionCapacity constrained service industries, such as trans-portation, tourism, entertainment, media, and internetproviders are constantly faced with the problem ofintelligently allocating their limited, perishable inven-tories to demand from different market segments,with the objective of maximizing total revenues. Rev-enue management is concerned with the theory andpractice underlying this type of problem. Followingairline deregulation, revenue management techniqueshave had an important impact on the developmentof the industry, providing up to 4%–10% increasesin company revenues (Fuchs 1987). For example, in1997, American Airlines collected one billion dollarsby implementing revenue management, representingmost of the company’s profit (Cook 1998).

Optimization techniques have been essential in thedevelopment of revenue management tools, particu-larly for seat allocation models. In this research, weare interested in investigating the design of dynamic

policies for allocating inventory to correlated, stochas-tic demand for multiple classes, in a network environ-ment. Specifically, we design a decision support tool,based on stochastic and dynamic optimization tech-niques, that at each point in time accepts or rejects areservation request, based on the currently availableinventory, past sales and future potential demand, soas to maximize total expected revenues.

Problem DefinitionThe main problem we address in this paper is as fol-lows. We are given an airline (hotel, car rental) net-work composed of l legs (pairs of consecutive daysfor hotels, car rentals), which are used to serve a totalof m demand classes. The initial inventory is given bya vector N = �N1� � � � �Nl� of leg capacities. The net-work is described by a l×m matrix A and a m-vectorR = �R1� � � � �Rm�: Rj is the fare category of class j ,which utilizes aij units of resource (leg) i. In this way,

0041-1655/03/3703/0257$05.001526-5447 electronic ISSN

Transportation Science © 2003 INFORMSVol. 37, No. 3, August 2003, pp. 257–277

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BERTSIMAS AND POPESCURevenue Management in a Dynamic Network Environment

a demand class j is defined by its itinerary Aj (a col-umn of matrix A) and its fare category Rj . In an airlinenetwork without group discounts, A is a 0–1 matrixwhich may contain repeated columns for each fareclass on a given itinerary. To account for special groupfares, integer multiples of the itinerary-incidence vec-tor are allowed.

For example, consider a very simple network cor-responding to a weekend in a hotel: There are threenodes Fri, Sat, Sun and l = 2 legs (1) Fri–Sat and(2) Sat–Sun with total capacities N = �N1�N2�. Sup-pose there is demand for all types of stays, i.e.,“itineraries:” (1) Fri–Sat, (2) Sat–Sun, and (3) Fri–Sat–Sun, with two (high and low) fare classes for each type.Moreover, suppose there are discounts for groups ofsize k1 = 10 for (1) Fri–Sat night stays, at a rate of R�10�

1

per group, that is a total of m = 7 classes. The leg-class incidence matrix, together with the correspond-ing fare structure R, is given by:

(RA

)=

Rh1 Rl1 Rh2 Rl2 Rh3 Rl3 R

�10�1

1 1 0 0 1 1 10

0 0 1 1 1 1 0

�We assume a finite booking horizon of length T ,

with the time line sufficiently discretized so as toallow at most one request (reservation or cancellation)per time period almost surely (a.s.). Time is countedbackwards: Time t = T is the beginning of the book-ing horizon and time t = 0 is the end of the reserva-tion period, where the no-shows are being counted.Customers who do not consume their reservations getfull, partial, or no refund, depending on their fareclass. If at the end of the horizon, the inventory isoversold, at time t = −1 redistribution decisions arebeing made. These can be class upgrades or customerbumping, in which case companies pay overbookingpenalties. For the case of hotels and car rentals, aninfinite time horizon could be more appropriate; how-ever, one can decompose the problem in fixed lengthtime periods (one month, one year, etc.).

The demand (to come) process at time t is denotedby �t , and ��t represents the corresponding ran-dom vector of cumulative demands. That is, ��t

j is arandom variable representing the number of class jrequests to come from time t until departure (order

does not matter). Usually, we have partial informationabout the demand process, which might consist of theexpected demand to come Dt = E��t�, and possiblyother type of information available from forecastingtools, such as cancellation or no-show probabilities.

The state of the system S is given by the time t(t periods to departure) and the sales-to-date records = �s1� � � � � sm� for each demand class. If cancellationsand no-shows are not allowed, then it is sufficient todefine the state based on the remaining inventory n =�n1� � � � �nl�. A common practice is to use the lattermodel and account for cancellations and no shows byincorporating a virtual increase, called overbookingpads, in the initial capacity definition.

The general stochastic, dynamic inventory controlproblem for network revenue management (NRM)can thus be stated as follows: At time t, and giventhat the state of the network is S, should we acceptor reject a new class j request? The overall objectiveis to maximize total expected revenues.

The decision to accept or reject determines anadmission control policy, which is in general a func-tion of the current network configuration (s or n),the time-to-go t, the currently requested fare Rj , aswell as partial information from demand forecasts. Ifwe accept the request, the new state becomes �s +kj · ej � t− 1�, where kj is the size of the class j grouprequest, and ej is the jth unit vector; when can-cellations are not allowed, the state of the network�n� t� becomes simply �n − Aj � t�. If the request isrejected, only the time component of the state vectoris changed to t−1.

In reality, the control policy has an impact onthe demand process, because customer choice maydepend on the opportunity set. Capturing and quan-tifying this type of feedback phenomenon is a sub-tle task that goes beyond the scope of this work. Wewill thus make the simplifying assumption that thedemand process is independent of the control policy.

NotationWe will use the following notation throughout thepaper. Vectors will be denoted in bold, and randomvariables and processes in calygrahic style.

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Time:T = length of time horizon (number of time peri-

ods);t = time periods left until departure (count-down).

Network:l = number of legs in the network;m = number of classes (itineraries with fare cate-

gories);N = total initial network capacity (l-vector);A = leg-class incidence �l×m�-matrix;aij = quantity of resource i utilized by bundle j .

Sales and inventory:n = remaining inventory (l-vector);s = sales to date vector (m-vector);stj = the number of itineraries sold to class j until

time t;soj = the number of class j itineraries overbooked at

departure.Fares, refunds, and penalties:Rj = revenue collected for one class j sold;Rcj = the refund per class j cancellation;Rnsj = the refund per class j no-show;Cj = the overbooking penalty per demand class j .

Demand:ptj = probability of request for class j at time t;pcjt = the probability of a class j cancellation occur-

ring at time t;pt0 = the probability of no request (reservation or

cancellation) at time t;pnsj = the probability that a class j reservation will

not show up;�t = demand (to come) process (m-dimensional);��t = aggregate demand (to come) distribution

(m-dimensional);Dt = E��t� = expected aggregate demand to come

(m-dimensional);� = the cancellation process, adapted to the sales

history (not a decision);�� = the no-show distribution (adapted to final

sales, adjusted for cancellations).We will use the operator �x�+ = max�x�0�, for

x ∈ R, which naturally extends for vectors: �x�+ =��x1�

+� � � � � �xn�+� for x = �x1� � � � � xn� ∈ Rn.

ContributionsIn this paper, we evaluate from different perspectivesseveral policies for solving the dynamic and stochastic

NRM problem. To compare these different policies,we provide structural and computational results.

The most popular technique developed in the cur-rent literature is an additive bid-pricing approach.These are mechanisms whereby the opportunity costof each itinerary is estimated as the sum of theshadow prices of the incident legs, obtained from alinear programming formulation of the problem (seeFormulation (2) in §2.2). However, there are two obvi-ous drawbacks to additive bid prices:

(a) They are not well defined if there are multipledual solutions.

(b) They are restrictive in their way of taking intoaccount bundles by their predefined additive struc-ture. In particular, they do not account for changes ofa dual basis in response to accepting large-group andmultileg itinerary requests.

The contributions of this paper are as follows:(1) We propose an efficient control policy, that is

well defined if there are multiple dual solutions,and does not have an additive structure. The pro-posed control policy, which we call certainty equiva-lent control (CEC), belongs in the class of approximatedynamic programming mechanisms (see Bertsekasand Tsitsiklis 1998), in which the cost-to-go function isapproximated by the value of a linear programming(LP) relaxation. We remark that this LP is equivalentto a network flow problem in the case of origin-destination (OD) demands or linear networks (wherenodes can be ordered so that the arcs are pairs of con-secutive nodes �i� i+1�� i = 1� � � � � l).

(2) We provide structural properties that com-pare the behavior of the proposed CEC policy withthe additive bid-price approach. These results offerinsight into the behavior of both methods.

(3) We propose several algorithmic improvementsof the CEC policy based on approximate dynamicprogramming.

(4) We provide computational results that giveinsight into the performance of these algorithms andseveral variations, for different networks and demandscenarios. We observe that the CEC algorithm per-forms very well in practice, giving results that arevery close to optimum. For high load factors, weobserve an average 5%–10% improvement over exist-ing policies (additive bid pricing). We describe and

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simulate extensions of this algorithm that result in sig-nificantly higher improvements (up to 20%). Interest-ingly, the CEC policy appears to be significantly morerobust to noise and bias in the demand forecast.

(5) We extend these algorithms to handle cancel-lations and no-shows by incorporating overbookingcontrol in the underlying mathematical programmingformulation. This extension preserves several struc-tural properties. We report computational results thatshow that the proposed algorithm improves upon theperformance of the bid-price control policy.

StructureThe remainder of the paper is organized as follows: Inthe next section, we present an overview of the litera-ture. Section 2 describes several (dynamic, stochastic,linear, and network flow) models and formulationsfor the NRM problem and evaluates the relationshipsbetween them. Section 3 presents efficient algorithmsfor the NRM problem based on ideas from approxi-mate dynamic programming. In §4, we present struc-tural properties for the proposed CEC policy andcontrast them with additive bid pricing. In §5, weextend our model and algorithms to handle cancella-tions, no-shows, and overbooking. Finally, in §6, wepresent computational results. The last section sum-marizes our conclusions.

1. Literature Reviewand Positioning

The NRM problem can be viewed as a particularinstance of the general class of perishable asset rev-enue management problems (PARM). Initial devel-opments in static single leg revenue managementare due to Littlewood (1972), followed by Simpson(1989) and Belobaba (1987), who proposed a subop-timal policy for computing protection levels basedon expected marginal seat revenues (EMSR). Curry(1989), Wollmer (1992), and Brumelle and McGill(1993), derive the optimal solution for the single legstatic model. Robinson (1995) proposes an extensionthat handles nonmonotonic fare classes.

A characterization of the optimal dynamic policybased on a threshold time property is due to Diamond

and Stone (1991). An analogous discrete time solu-tion is provided by Lee and Hersh (1993), whoalso provide a method for estimating arrival rates.Subramanian et al. (1999) propose a dynamic pro-gramming model that handles cancellations and over-booking, by analogy to a problem in the optimalcontrol of admission in a queueing system.

A natural, but much harder question is to deter-mine which of the single leg results can be extendedto network (multiproduct) settings, and how. A majorconceptual advance in the study and practice of net-work revenue management was introduced by bid-price control. These are additive, leg-based shadowprices used to approximate the opportunity cost ofitinerary capacity. The concept was proposed bySimpson (1989), and further analyzed by Williamson(1992) in extensive simulation studies. A deficiency ofsuch mathematical programming based models is thatthey do not account for nesting of the fare classes.To overcome this problem, Curry (1992) proposes avirtual nesting method. In all cases, however, the allo-cation of capacity to itinerary demand is decided bya one-time, static rule (one fixed set of bid prices).

The most realistic and relevant, yet least investi-gated model for NRM is the dynamic network model.Talluri and van Ryzin (1998) study a dynamic net-work model using bid-price control mechanisms andargue why bid-price policies are not optimal in gen-eral. They provide an asymptotic regime when certainbid-price controls, based on a probabilistic program-ming formulation of the problem, are asymptoti-cally optimal. In the context of hotels, Bitran andMondschein (1995) propose a dynamic policy thatextends to multiple-night stays, but do not give anyfurther analysis.

Chen et al. (2000) formulate the problem as aMarkov decision problem, and use linear program-ming and regression splines to approximate the valuefunction. Gunther et al. (2000) introduce a newmethod to compute bid price for single hub airlinenetworks. Both studies report encouraging simulationresults. None of these take into account cancellations.

For further pointers to related literature, we referthe interested reader the survey of McGill and


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van Ryzin (1999), and for a schematic summary thePh.D. thesis of Popescu (1999).

Relative Positioning. Our approach can be viewedas extending the single-leg models investigated byLee and Hersh (1993) and Bitran and Mondschein(1995), who actually propose a similar LP-basedheuristic for the multiple-night booking problem ina hotel, but without any further analysis. Our over-booking model is similar to the early single-leg DPformulation of Rothstein (1971, 1974). As far as staticnetwork models are concerned, our LP formulation issimilar to the one proposed by Williamson (1992), butwe handle multiple classes and group bookings. Thenetwork flow formulation we provide for linear net-works and origin-destination fare (ODF) demands, isthe same as those proposed by Glover et al. (1982) forairlines and Chen (1998) for hotels.

2. General Modelsand Relationships

The problem of dynamic inventory control for NRMbelongs in the class of finite horizon decision prob-lems under uncertainty. In this section, we presentseveral optimization models for addressing the NRMproblem, and discuss various relationships betweenthem. Again, we assume that the control policy doesnot feed back into the evolution of the demand pro-cess. We restrict to the case when cancellations andoverbooking are not permitted; this case is discussedin detail in §5.

2.1. The Dynamic Programming ModelThe stochastic dynamic programming model providesthe optimal policy for the NRM problem, by evaluat-ing the whole tree of possibilities and making at eachpoint in time the decision (to sell or not to sell) thatwould imply higher future expected revenues.

The states S = �n� t� are defined by the currentavailable capacity vector n when the remaining timeis t. The stochasticity is given by the demand processto come �t for the remaining t periods. We defineDP�n� t� to be the maximum expected revenue tobe collected from state �n� t�. Assuming independent

demands, this can be computed via the Bellman equa-tion as follows:

DP�n� t�

=m∑j=1

ptj max�DP�n� t−1��Rj +DP�n−Aj � t−1��

+pt0DP�n� t−1�

= DP�n� t−1�+m∑j=1

ptj(Rj −DP�n� t−1�

+DP�n−Aj � t−1�)+ (1)

for all n ≤ N , t ≤ T , with the boundary conditions:

DP�n� t�=− if ni < 0 for some i� and

DP�n�0�= 0� for n ≥ 0�

We define the opportunity cost OCj �n� t� to be

OCj �n� t�= DP�n� t−1�−DP�n−Aj � t−1��

Then, the optimal policy accepts a request if and onlyif the corresponding fare Rj exceeds its current oppor-tunity cost. The value function can thus be expressedas follows:

DP�n� t�= DP�n� t−1�+m∑j=1

ptj �Rj −OCj �n� t��+�

One can easily show that the value function is non-decreasing in n and t. Moreover, for the single legcase without batch arrivals, the value function is con-cave and opportunity costs decrease with t and n (seeDiamond and Stone 1991). Based on these properties,one can show that the optimal policy is characterizedby threshold times. Threshold times are points in timeduring the booking horizon before which requests arerejected, and after which requests are accepted. Inthe general network case, however, this is not true, aswe will show in an example in §4.1.

2.2. The Integer and Linear ProgrammingModel (IP, LP)

The most frequently utilized formulations for theNRM problem are static models. These are determin-istic analogues of the stochastic dynamic problem,


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that use only expected demand information, and areusually much simpler to solve.

Suppose that all the available demand informationconsists of (unbiased) forecasts of the expected aggre-gate demand to come Dt = E��t� over the remainingt periods. The integer programming model computesthe optimal allocation y∗ of available inventory nto the expected itinerary demand Dt , by maximiz-ing total revenues subject to capacity and itinerarydemand constraints. For all n ≤ N and t ≤ T , we have

IP�n�Dt� = max R′ ·ys.t. A ·y ≤ n

0 ≤ y ≤ Dt

y integer�

The linear programming relaxation of this problemprovides an efficient way to compute the best possible“fractional” allocation of inventory, and is defined bysimply relaxing the integrality constraint:

LP�n�Dt� = max R′ ·ys.t. A ·y ≤ n

0 ≤ y ≤ Dt � (2)

Clearly, we have that IP�n�Dt� ≤ LP�n�Dt�. Thisinequality can be strict, but there are particularinstances when equality holds, one of which (linearnetworks) we describe in the next section. One canobtain further insight into the optimal LP-allocationby looking at the dual problem:

LP�n�Dt� = min v′ ·n+u′ ·Dt

s.t. v′ ·A+u′ ≥ R′

u�v ≥ 0�

This formulation can be equivalently written as:

LP�n�Dt� = minv≥0

v′ ·n+ �R′ −v′ ·A�+ ·Dt

= mink∈K

v′k ·n+u′

k ·Dt�

where K denotes the index set of extreme points�vk�uk� of the dual polyhedron. Thus, the objectivevalue of Model (2) is a piecewise linear, concave, andnondecreasing function of the expected demand tocome Dt and available capacity n. We next relate theobjective values of the dynamic and static formula-tions at any state �n� t�.

Proposition 1. DP�n� t�≤ LP�n�Dt�.

For a full proof see Popescu (1999). The idea isthat for each pathwise realization of demand, the rev-enue collected by the DP-policy along that path isupper bounded by the value of a “perfect hindsight”stochastic program. The perfect hindsight model,denoted PI, determines the optimal allocation ofinventory for each particular realization of demand,and then computes the expected reward over allscenarios:

DP�n� t� ≤ PI�n��t�= E��LP�n��t��

≤ LP�n�E��t��= LP�n�Dt��

Here, �t� is the cumulative demand corresponding

to scenario �. Since the LP value is concave in thedemand vector, the second inequality follows fromJensen’s inequality. Furthermore, the two values con-verge as demand becomes very large (see Popescu1999).

2.3. Origin-Destination RM andLinear Networks

Consider a particular case of the NRM problem,where demand is by ODF, as opposed to itineraryspecific. This is by default the case for linear net-works, where there is no question of routing, such asin hotel or car rental revenue management (where thenodes are days). In the case of airline revenue man-agement this situation occurs when there are perfectlysubstitutable routes (same price, same travel time, andso on). In these cases, the Model (2) can be repre-sented as a network flow problem. The advantage ofthis formulation is that it is very easy to solve in prac-tice and reoptimization is very fast. For airlines, thismodel provides (for the company) optimal routing ofpath-indifferent customers.

The network representation is as follows: The nodesare the origins and destinations (days for hotels, {air-port, time}-pairs for airlines). There are multiple for-ward arcs �o�d�f representing the flow from origin oto destination d from fare-class f . The capacity of eachforward arc is the corresponding aggregate demandDf

od. The revenue collected along arc �o�d�f is Rfod. Foreach “leg” �i� j� (flight leg, respectively, pair of consec-utive days) there is a backward arc of the type �j� i�,


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capacitated by nij , the available inventory of leg �i� j�.The revenue collected along backward arcs is zero.This model has been proposed by Glover et al. (1982)in the context of airlines, and for hotel revenue man-agement by Chen (1998), who proves that it reducesto solving a network flow problem.

Proposition 2 (Integrality of Solution). ForODF models with integer data, the optimal LP-solution isintegral, that is IP�n�Dt�= LP�n�Dt�.

3. Approximate DynamicProgramming Algorithms

In this section, we present several algorithms for theNRM problem. Most of these algorithms belong in thegeneric class of approximate dynamic programmingmethods (see Bertsekas and Tsitsiklis 1998), in whichan approximate value to the exact value function isused in the Bellman equation.

Given a certain efficient mathematical program-ming formulation (MP) of the NRM problem, ageneric approximate DP algorithm for the NRM prob-lem has the following structure:

Generic MP-Policy. Identify an efficient formula-tion MP of the NRM problem.At any current state S = �n� t�,

(1) For a class j request, compute an MP-based esti-mate of the opportunity cost OCMP

j �S�.(2) Sell to class j if and only if its fare Rj exceeds

its opportunity cost estimate, i.e.,

Rj ≥ OCMPj �S��

(3) Go to Step 1 and ITERATE.We will denote the expected value of this policy at

an initial state S as MP�S�. The difference betweenvarious algorithms comes from the approximating MPand the MP-based opportunity cost measure, that isfrom Step 1.

3.1. Bid-Price ControlBid-price control is a popular method in NRM,whereby the opportunity cost (shadow price, or bidprice) of an itinerary is approximated by the sum ofopportunity costs of the legs along that itinerary. First,opportunity cost estimates (bid prices) are determinedfor each leg in the network, usually as the leg-shadow

prices v from the linear programming formulation (2).Then itinerary bid prices are computed additively, ateach state S = �n� t�, and for each class j request as

BPj �S�= �vS�′ ·Aj � (3)

Notice that bid prices depend on the choice ofoptimal dual variables vS. This technique was ini-tially proposed by Simpson (1989), then studied byWilliamson (1992) in her Ph.D. thesis. Several proba-bilistic models have been investigated by Glover et al.(1982), Williamson (1992), and Talluri and van Ryzin(1998). It has been observed (Williamson 1992) thatthe LP model achieves a better performance, and ismore efficient. For recent work on bid-price controlsee Talluri and van Ryzin (1998) and Gunther et al.(2000).

3.2. Certainty Equivalent ControlThe main disadvantages that are apparent from thedefinition of leg-based additive bid prices is that(a) they are not uniquely defined (several sets ofshadow prices may be optimal), and (b) they providean additive approximation of the opportunity costs,which are not necessarily additive due to “bundleeffects” (group or multileg itinerary requests maydetermine basis changes in the dual LP).

We provide a different approximation scheme foropportunity costs, based on certainty equivalentadaptive control. The idea is to approximate the valuefunction of the dynamic program DP�n� t� defined inEquation (1) by the value of the linear programmingproblem LP�n�Dt� defined in §2.2. Thus, in Step 1 ofthe generic algorithm, a request for a given class jwill be accepted if and only if its price Rj exceeds itscurrent opportunity cost estimate given by:

OCLPj �n� t�= LP�n�Dt−1�−LP�n−Aj �Dt−1��

Because the opportunity cost estimate is calculatedin terms of LP objectives, it is uniquely defined, inthat its value will not depend on the choice of (dual)solution. Thereby, the first drawback of bid prices isresolved.

This is the certainty equivalent control (CEC) pol-icy, and we denote its expected value for an initialstate S as LP�S� = CEC�S�. For more on certainty


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equivalence, see Bertsekas (1995). A similar policy isproposed by Bitran and Mondschein (1995) in the con-text of hotel revenue management, but no analysis isprovided. Notice that in the case of linear networks,it is actually desirable to use the equivalent networkflow formulation as described in §2.3 instead of theusual LP model because reoptimization at each stagebecomes a much simpler task.

More generally, one can use this technique with vir-tually any mathematical programming (MP) modelthat provides an approximation of the value func-tion. The corresponding OC estimate for an itinerary jrequest at the current state S is defined as OCMP

j �S�=MP�Srej�j��−MP�Sacc�j��, where Sacc�j� and Srej�j� are thestates corresponding to the accept, and respectivelyreject decision for the current request j . For exam-ple, for the static LP model without cancellations andoverbooking, if S = �n� t�, then Sacc�j� = �n−Aj � t− 1�and Srej�j� = �n� t−1�.

3.3. ExtensionsIn this section, we describe several extensions thatprovide improvements in the efficiency or accuracy ofthe adaptive policies described above.

Rollout Policy. The expected value H�S� of anyheuristic H started in state S provides an approxi-mation of the DP�S� value. Therefore, the expectedheuristic values can in turn be used to provide esti-mates of opportunity costs, determined as follows:

OCR�H�j �n� t�= H�n� t−1�− H�n−Aj � t−1��

This opportunity cost estimation mechanism leadsto a new approximate dynamic programming (ADP)heuristic, called the rollout of H , and denoted R�H�.This method is simply a form of policy iteration andis described in detail in Bertsekas and Tsitsiklis (1998).It has been observed in the dynamic programmingliterature that this procedure systematically improvesheuristic performance (see also Bertsimas et al. 1999,Bertsekas et al. 1997, Bertsekas and Castanon 1998).

For practical purposes, we suggest using MonteCarlo simulation for evaluating the policy value Hfor a subset of states, and then interpolating these inan online fashion. An interesting research idea is toinvestigate what types of preprocessing simulationswould provide an insightful information database.

Simulations Using Monte Carlo Demand Esti-mation. One problem with the certainty equivalentpolicy is that it only considers expected demand infor-mation, and uses a deterministic approach to a highlystochastic problem. We propose a variation on thecertainty equivalent policy that uses Monte Carlodemand estimation to capture demand variability.

Suppose we have information that the cumula-tive demand to come ��t−1 follows a certain distribu-tion. We generate r samples from this distribution:Dt−1

1 � � � � � Dt−1r . In Step 1 of the generic algorithm, we

calculate the opportunity cost estimate of itinerary jas a weighted average,

OCMCj �n� t�=

r∑i=1

$i ·OCLPj �n� Dt−1

i ��

where $i = P��t−1 = Dt−1i ��t−1 ∈ &Dt−1

1 � � � � � Dt−1r '�

and OCLPj �n� Dt−1

i �= LP�n� Dt−1i �−LP�n−Aj � Dt−1

i �.One difficulty with implementing this procedure is

that we might not have enough information about theaggregate demand and/or it may be too expensive tocompute the actual value of the conditional probabil-ities $i. For this reason, we run a simplified versionof this policy, that assigns the same weights to all thetrials: OCMC

j �n� t�= �1/r� ·∑ri=1 OCLP

j �n� Dt−1i �.

4. Structural PropertiesIn this section, we derive several structural propertiesof the new approximate dynamic programming algo-rithm (CEC) and compare it with additive bid-pricecontrol algorithms (BPC) developed in the literature.

Given that the NRM problem requires a real-time response, it is desirable to use computationallyinexpensive models to construct approximations ofthe opportunity cost. This motivates the choice forusing the LP formulation described in §2.2. We areinterested in a comparative structural assessment ofthe two LP-based policies described previously, BPCand CEC, and their corresponding opportunity costapproximations.

From an asymptotic point of view, it should benoted that the CEC policy is asymptotically optimalin the fluid scaling regime proposed by Talluri andvan Ryzin (1998), whereby demand and capacitiesare simultaneously increased in a way that preserves


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their relative values constant. They prove that in thisregime, the additive bid-pricing policy converges tothe optimum, as bid prices are being held fixed. Byimitating their proof, one can show that the sameproperty holds for the CEC policy, with determinis-tic prices, as OC estimates are being held fixed (seePopescu 1999).

The next result compares the opportunity costapproximations of a class j request at any given state.

Proposition 3. In any state �n� t�, for any bid pricesBPj �n� t��BPj �n−Aj � t�, the following inequalities hold:

BPj �n� t�≤ OCLPj �n� t�≤ BPj �n−Aj � t�� (4)

Inequalities are strict if accepting class j must incur achange of basis in the LP dual.

Proof. Recall that we have defined:

OCLPj �n� t� = LP�n�Dt−1�−LP�n−Aj �Dt−1�

= �vn� t�′ ·n+ �un� t�′ ·Dt−1

− �vn−Aj � t�′ · �n−Aj �− �un−Aj � t�′ ·Dt−1�

where �vn� t�un� t� and �vn−Aj � t�un−Aj � t� are optimaldual solutions of LP�n�Dt−1� and LP�n − Aj �Dt−1�,corresponding to the given bid prices: BPj �n� t� =�vn� t�′ · Aj and BPj �n − Aj � t� = �vn−Aj � t�′ · Aj , respec-tively. Because both solutions are feasible for bothprograms, we obtain the following upper bounds byevaluating each LP at the optimal solution of theother:

LP�n�Dt−1�≤ �vn−Aj � t�′ ·n+ �un−Aj � t�′ ·Dt−1�

LP�n−Aj �Dt−1�≤ �vn� t�′ · �n−Aj �+ �un� t�′ ·Dt−1�

The upper bound in Equation (4) follows by applyingthe first of these inequalities in the OC formula, andthe lower bound by the second inequality,

BPj �n� t� = �vn� t�′ ·Aj ≤ OCLPj �n� t�≤ �vn−Aj � t�′ ·Aj

= BPj �n−Aj � t�� (5)

In case the two dual optimal solutions coincide, weobtain equality throughout. �

In general, if at a given state the CEC policy acceptsa class j request, then at the same state, the bid-pricing

Table 1 The Behavior of the BPC and CEC Policies as a Function of anOptimal Primal Solution y∗

y ∗j ≥min�Dj �1� y ∗

j <min�Dj �1� in all y∗,y ∗j in some y∗ but �= 0 in some y ∗

j = 0 in all y∗

CEC accept reject rejectBPC accept accept reject

policy will also accept, but not vice versa. This isbecause the following situation may occur: BPj �n� t�≤Rj <OCLP

j �n� t�, (see §4.1).Table 1 provides a comparative characterization of

the bid-pricing policy (BPC) versus the CEC policy, interms of the structure of the primal optimal solutionsy∗ of the LP model (2). We assume the BPC policyis well defined, in that the dual optimal solution isunique. We also assume that the dual basis is not thesame for LP�n�Dt−1� and LP�n−Aj �Dt−1�; if the dualbasis does not change, then the policies are identical.The following two propositions state and prove theseresults formally.

Proposition 4 (Structural Properties of theBPC Policy). At any state �n� t�, if LP�n�Dt−1� hasa unique dual optimal solution, then the correspondingbid-price policy accepts only classes j for which y∗j > 0 insome primal optimal solution.

Proof. By analyzing the primal and dual LP, onecan distinguish the following situations:

• In all optimal LP-solutions y∗j = 0, then un� tj = 0

from complementary slackness. By strict complemen-tary slackness (see Bertsimas and Tsitsiklis 1997,p. 192), we have �vn� t�′ ·Aj+un� t

j > Rj , i.e., BPj �n� t�=�vn� t�′ · Aj > Rj , in which case the bid-price policyrejects class j .

• In all optimal LP-solutions y∗j = Dt−1j , in which

case un� tj = �R′

j − �vn� t�′ ·Aj �+ > 0, so the bid-price pol-icy accepts class j .

• There is some optimal LP-solution such that 0<y∗j < D

t−1j , which implies that the dual constraint

�vn� t�′ · Aj − un� tj ≥ Rj is binding and un� t

j = 0, so�vn� t�′ ·Aj = Rj , and thus the bid-price policy acceptsclass j . �

Proposition 5 (Structural Properties of theCEC Policy). Suppose that LP�n − Aj �Dt−1� andLP�n�Dt−1� have different optimal dual bases. Then:


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(a) CEC accepts class j if y∗j ≥ min�Dt−1j �1� in some

optimal solution of LP�n�Dt−1�.(b) CEC rejects class j if y∗j <min�Dt−1

j �1� in all opti-mal solutions of LP�n�Dt−1�.

Proof. If in some optimal solution y∗ of LP�n�Dt−1� we have that y∗j ≥ min�Dt−1

j �1�, then y∗ −min�Dt−1

j �1� ·ej ≥ 0 is a feasible solution of LP�n−Aj �

Dt−1�j� �, where Dt−1

�j� =Dt−1−min�Dt−1j �1� ·ej , and hence,

LP�n�Dt−1� = R′ ·y∗≤min�Dt−1j �1�Rj+LP�n−Aj �Dt−1

�j� �

≤ Rj+LP�n−Aj �Dt−1��

where the last inequality holds because any opti-mal primal solution of LP�n − Aj �Dt−1

�j� � is feasibleto LP�n −Aj �Dt−1�. So, OCLPj �n�Dt� = LP�n�Dt−1�−LP�n−Aj �Dt−1�≤ Rj . This proves Part (a).

For Part (b), it follows that 0 ≤ y∗j < Dj in all pri-mal optimal solutions. By complementary slackness,we have that un� t

j = 0 and �vn� t�′ ·Aj = Rj . Under theassumption that the optimal dual basis changes, weobtain by Proposition 3 that OCLP

j �n� t� > BPj �n� t� =�vn� t�′ ·Aj = Rj , that is CEC rejects class j , which con-cludes the proof. �

4.1. An ExampleFor single-leg instances of the NRM problem, the bidpricing and the CEC algorithms are the same. Thisis because under the CEC algorithm, the opportu-nity cost estimates are the same (no change of basisoccurs in Equation 4). However, this is not true forthe general network case. In this section, we providean example that highlights the differences betweenthe BPC and the CEC policies, and shows instanceswhere each one is suboptimal. Moreover, we explainwhy the cross-concavity properties that insure thethreshold time structure for the optimal single-legpolicy cannot be extended to the network case. Wewill use the same example to exhibit the followingsituations:

• An instance when BPC accepts, but CEC rejectsa given request in the same state;

• An instance when BPC is suboptimal;• An instance when CEC is suboptimal;• A counterexample of a cross-concavity property

(“decreasing differences”) of the LP and DP-valuefunctions for the NRM problem.

In general, if at a given state the CEC policy acceptsa request from itinerary j , then at the same state thebid-pricing control policy will also accept, but not viceversa (see Proposition 3). The following situation mayoccur: BP�j� ≤ Rj < OCLP�j�, and so we will acceptunder the BPC policy but not under the CEC policy.The following is an example of such behavior, thatprovides insight into the structural properties of thetwo policies.

Consider a network with four nodes: a hub h, twoorigin nodes o1� o2, and a destination node d. The legsof the network are (1) �o1�h�, (2) �o2�h�, (3) �h�d�. Onecan think of this as part of a bigger network wherethe other (connecting) flights have been sold out. Sup-pose there is demand from the origin nodes o1� o2 tothe hub node h and to the destination node d, on theitineraries: (1) o1h, (2) o2h, (13) o1hd, (23) o2hd. Sup-pose that there is only one fare class per itinerary,and there is no demand from h to d. We assume thepricing structure is such that subitineraries cost less:R1 < R13�R2 < R23. Furthermore, assume with almostno loss of generality that R13+R2 <R23+R1 (the othercase is symmetric, unless equality holds).

Suppose that the available capacity in the currentstate t is n = �n1�n2�n3� and the expected demandfor each fare class in the remaining t− 1 periods ispositive. The static LP and its dual can be formulatedas follows:

LP�n�D� = max R1y1 +R2y2 +R13y13 +R23y23

s.t. y1 +y13 ≤ n1

y2 +y23 ≤ n2

y13 +y23 ≤ n3

0 ≤ y ≤ D

= min n′v+u′Ds.t. v1 +u1 ≥ R1

v2 +u2 ≥ R2

v1 +v3 +u13 ≥ R13

v2 +v3 +u23 ≥ R23

u�v ≥ 0�

Suppose that there is one seat left on each leg, so n =�1�1�1�, and D1 > 1 and D23 > 1, so that the demandfor the high-paying mix is large enough for the corre-sponding constraints to be nonbinding in an optimalsolution. Then the optimal LP solution is y∗1 = y∗23 = 1,y∗2 = y∗13 = 0, and the value of the LP is R1 +R23.


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Since the demand constraints are nonbinding, wemust have that u = 0, and hence v1 ≥R1�v2 ≥R2�v3 ≥max�0�R13 − v1�R23 − v2�. Therefore, in an optimalsolution, the shadow prices are equal to v∗1 = R1�v

∗2 =

R2�v∗3 = R23 −R2. We can compute the bid prices and

opportunity costs as follows:

OCLP1 = LP�1�1�1�D�−LP�0�1�1�D�

= R1 = BP�1��OCLP

2 = LP�1�1�1�D�−LP�1�0�1�D�= R1 +R23 −R13 > BP�2�= R2�

OCLP13 = LP�1�1�1�D�−LP�0�1�0�D�

= R1 +R23 −R2 = BP�13� > R13�

OCLP23 = LP�1�1�1�D�−LP�1�0�0�D�

= R23 = BP�23��

Therefore, the two policies disagree on the accep-tance of class (2): Under the BPC policy, we willaccept, whereas under the CEC policy, we will rejecta class (2) request at state n = �1�1�1� as long asthere is sufficient forthcoming demand for classes (1)and (23).

The question is which one of the policies is better?Clearly, in the case when demand is deterministic,the BPC policy is suboptimal, by giving away at timet one unit of capacity that would bring higher rev-enues in the future. The CEC policy, however, is bydefinition optimal in the deterministic case becauseit is equivalent to the DP (certainty equivalence). Inthe stochastic case, the bid-pricing policy is subopti-mal when there is sufficient demand to come from thehigh-fare mix. Otherwise, we may be better off accept-ing class (2) right away, in which case our policy issuboptimal.

We can also observe on this example that the LP,and thus the DP value do not exhibit a certain type ofcross-concavity property called decreasing differences(see Karaesman and van Ryzin 1998):

Definition 1. A function f . S ⊂ Rn → R satisfiesdecreasing differences on S if for any s ∈ S, and i �= j ∈&1� � � � �n', and for all 0i�0j > 0 with s+ 0iei, s+0jejand s + 0iei + 0jej ∈ S, the following relation holds:f �s+0iei+0jej �− f �s+0iei�≤ f �s+0jej �− f �s�.

This cross-concavity property reduces in the uni-variate case to concavity. This is the key observationunderlying the proof of the threshold times propertyfor the optimal single-leg policy (see Diamond and

Stone 1991), and would provide a sufficient conditionfor the property to extend to the network case.

However, the decreasing differences property isviolated in our example:

LP�1�1�1�−LP�0�1�1�

= max�R1 +R23�R2 +R13�−R23 <R13

= LP�1�0�1�−LP�0�0�1��

Notice that the assumption here is that“subitinerary” fares are cheaper (R1 < R13�R2 < R23).Network effects imply that the opportunity cost ofItinerary (13) under the CEC policy decreases withcapacity. Furthermore, if the demand is large enough,the above relation transfers to the corresponding DPvalues because the two are asymptotically equal.Surprisingly, this says that incremental revenues(opportunity costs) may decrease by decreasingcapacity along a certain direction (see also Feng andLin 2000).

5. Cancellations and OverbookingThe control policies we have discussed so far donot take into account the fact that a significant frac-tion of customers cancel their reservations duringthe booking period (cancellations) or simply do notutilize their reservation (no-shows). In these casescustomers get full, partial, or no refund, dependingon the fare category. In either case, extra capacitybecomes available and could be used to accommodateother potential customers. To counterbalance this phe-nomenon, a common revenue management practice isoverbooking. Airlines, hotels, and so on, oversell theirinventories to account for reservations that will notmaterialize. If at the end of the horizon, more demandhas materialized than the inventory can accommo-date, companies practice class upgrades, pay over-booking penalties to unsatisfied reservations or evenperform aircraft changes.

The typical overbooking method practiced by air-lines, hotels, and so on, is to decide an initial alloca-tion of overbooking pads, which are virtual increasesin leg-capacity. This is usually performed, in prac-tice as well as in the literature, as a static, one-time


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decision made at the beginning of the booking hori-zon. In the following, we propose a new methodfor dynamic overbooking control, where oversalesdecisions are dynamic and implicit in the admissioncontrol mechanism.

5.1. General Models and RelationshipsFirst, we extend the various models described in §2 toincorporate cancellations, no shows and overbooking.

A Dynamic Programming Model. In the case ofperfect state information, this model provides the opti-mal control policy. By allowing cancellations, the statespace of the DP-model becomes much larger becauseit is necessary to keep track of the past sales record s.The random quantities involved are the demand, can-cellations, and no-show processes. Given the initialnetwork inventory N, the maximum expected rev-enue (less refunds and penalties) to be collected fromstate �s� t� onward (cost-to-go), is given by

DPoN�s� t� =∑j

ptj ·max(DPoN�s� t−1��

Rj +DPoN�s+ej � t−1�)

+ ∑j sj≥1

pctj · �DPoN�s−ej � t−1�−Rctj �

+pt0 ·DPoN�s� t−1�

DPoN�s�0� = ∑s

P�s bookings out of s materialize�

·DPoN�s�−1�

DPoN�s�−1� = −min C ′ · sos.t. A · �s− so�≤ N

0 ≤ so ≤ s� (6)

The difference from the basic model is that cancel-lation events are incorporated in the Bellman Equa-tion (6), and the boundary conditions are changed toaccount for no-shows (time t = 0) and final bumpingdecisions (time t =−1). The final bumping decision ismade so as to minimize total penalties, while keepingthe actual capacity restrictions satisfied.

If customer-walking penalties are payed per legc = �c1� � � � � cl�, rather than per itinerary, the bound-ary condition at t = −1 is simply c′�As−N�+. Whenbumping penalties are itinerary specific (not leg-additive), then an optimization problem needs to be

solved to decide which passengers should be refusedboarding so as to incur least penalties. For example,in a two-leg network which is oversold by one seaton each leg, it is better to bump a connecting passen-ger rather than two different passengers on each legwhenever overbooking penalties are leg-subadditive.

This model does not incorporate secondary effectsassociated with overbooking (e.g., image damage andloss of goodwill, customer value, demand shifting,and so on).

The Integer and Linear Programming Models. Inthe case of the NRM problem with cancellations andno shows, we observe that both the final revenuegained from, and capacity occupied by a class j reser-vation are not deterministic quantities because theydepend on cancellations and no-shows which are ran-dom processes. To define a model that is consistentwith these observations, we define Rj and Aj to be theexpected revenue gained from, and expected capacityoccupied by a class j reservation, before the overbook-ing period. Let pcj and pnsj denote the probability that agiven class j reservation is cancelled at some point inthe booking period, and respectively does not showup for the flight. We assume that these quantities areindependent on the time the reservation was made,and so is the cancellation penalty. We denote the rev-enue expected (or “adjusted”) from booking a classj customer as Rj = Rj − �1− pnsj � · pcj ·Rcj − �1− pcj � · pnsj ·Rnsj , which accounts for potential cancellation and no-shows events and respective refunds Rcj �R

nsj (but not

overbooking penalties). Let Aj = �1− pcj � · �1− pnsj � ·Aj

denote the expected capacity occupied by a class jreservation at the end of the horizon. Finally Cj =�1− pcj � · �1− pnsj � ·Cj is the average overbooking costof one class j reservation.

With these notations, we can formulate the fol-lowing integer programming approximation model,that maximizes expected revenues subject to expectedcapacity constraints:

IPoN�s� t� = max R′ ·y−C′ ·zos.t. 0 ≤ A · �y+ s�−A ·zo ≤ N

0 ≤ y ≤ Dt

y�zo integer�

The vector y decides how many requests to beaccepted in the future, whereas zo determines which


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customers (itinerary requests) should be bumped atthe end of the horizon, if necessary. The “virtual”overbooking pad for each leg is A · zo. The capacityconstraint requires that cancellation-adjusted past andfuture sales, less oversales, should not exceed the ini-tial network capacity.

With the change of variable soj = zoj /�1−pcj � ·�1−pnsj �,j = 1� � � � �n, the corresponding linear programmingrelaxation, and its dual, can be written as follows:

LPoN�s� t� = max R′ ·y− C′ · sos.t. A · �s+y− so�≤ N

0 ≤ y ≤ Dt

0 ≤ so ≤ y+ s

= min v′ ·N−u′ · s+ �R′ −u′�+ ·Dt

s.t. u′ = min�C′�v′ · A�v ≥ 0�

5.2. Adjusted PoliciesFollowing the spirit of the basic adaptive bid pricingand CEC policies, we adjust these to incorporate can-cellations and overbooking.

Adjusted BPC Policy. We modify the DP model sothat at any given state S a class j request is accepted ifand only if its “adjusted” fare Rj is higher than eitherits adjusted bid price, or the adjusted overbookingpenalty Cj , i.e.,

Rj ≥ min�Cj�BPoj �S��

Here, bid prices are computed as BPoj �S� = �vS�′ · Aj ,where vS represent shadow prices for the leg-capacityconstraints in LPo�S�.

Adjusted CEC Policy. Similarly, we can adapt theCEC policy to accept a class j request if and only ifits “adjusted” revenue exceeds its “adjusted” oppor-tunity cost. Rj≥OCoj �S�= LPoN�s� t−1�−LPoN�s+ej � t−1�. Again, adjustment accounts for the fact that capac-ity and revenue might not be realized or may beoverbooked.

The same type of arguments can be used to extendProposition 1 for the case of cancellations, no showsand overbooking. Moreover, the structural proper-ties proved in §4 are preserved in the overbooking-adjusted policies.

6. Computational ResultsIn this section, we present computational results thatillustrate the relative practical performance of thepreviously described admission control policies forthe NRM problem, under different demand regimes.Our objective is to evaluate the different models andpolicies proposed, in terms of the following criteria:(1) running time, (2) quality of approximation, and(3) robustness. In addition, we would like to fur-ther investigate the effectiveness of several extensionsand improvements. In §6.1, we report computationalresults without cancellations and overbooking, whilein §6.2, we allow cancellations and overbooking.

6.1. Computational Performance WithoutCancellations and Overbooking

We performed expected value calculations for two-leginstances and simulation runs for larger networks. Tohave a consistent and fairly accurate base for compar-ing various policies, these were simulated simultane-ously on the same realizations of the demand process.The computations were performed in MATLAB 4.0 onan Intel Pentium II Celeron 450 MHz (128 MB RAM,WinNT 4.0). We restricted our attention to smallerinstances with the explicit objective to obtain insightsinto the behavior of both BPC and CEC.

6.1.1. Models and Assumptions. We consideredthe following types of networks:N2. A two-leg network with nodes o�h�d, legs

(1) oh (2) hd and capacities N = �N1�N2�. The availableitineraries are (1) oh, (2) hd, and (3) ohd, with one classper itinerary. The leg-class incidence matrix, togetherwith the fare structure R, is:

(R

A

)=

R1 R2 R3

1 0 1

0 1 1

�N3. The three-leg network described in the exam-

ple of §4.1, with four nodes: a hub h, two origin nodeso1� o2, and a destination node d, and the legs (1) o1h,(2) o2h, (3) hd. There is demand from the origin nodes


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o1� o2 to the hub node h and to the destination node d,on the itineraries: (1) o1h, (2) o2h, (13) o1hd, (23) o2hd.Suppose that there is only one fare class per itinerary,and there is no demand from h to d. The leg-class inci-dence matrix, together with the fare structure R, is:

(R

A

)=

R1 R2 R13 R23

1 0 1 0

0 1 0 1

0 0 1 1

�

N4. A four-leg network, with two origins o1� o2,two destinations d1�d2 and a hub h. The legs in thenetwork are (1) o1h, (2) o2h, (3) hd1, (4) hd2, with capac-ities N = �N1�N2�N3�N4�. There is demand for allthe eight itineraries, with one fare class per itinerary.The leg-class incidence matrix, together with the farestructure R, is:

(R

A

)=

R1 R2 R3 R4 R13 R14 R23 R24

1 0 0 0 1 1 0 0

0 1 0 0 0 0 1 1

0 0 1 0 1 0 1 0

0 0 0 1 0 1 0 1

�

N4�2. The same as �N4�, except there are two faresper itinerary, so 16 demand classes in all. The leg-class incidence matrix, together with a high-low farestructure R = �Rh�Rl�, is:

(R

A

)=

Rl1 Rh1 Rl2 Rh2 Rl3 Rh3 Rl4 Rh4 Rl13 Rh13 Rl14 Rh14 Rl23 Rh23 Rl24 Rh24

1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0

0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1

0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0

0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1

�

We used the following alternative scenarios tomodel the arrival process:HP. Homogeneous Poisson arrivals with arrival

rates given by a constant vector p (i.i.d. Bernoullitrials).NHL. Nonhomogeneous Poisson with high-low

demand. We assume arrival rates increase for high-fare classes (pt = p/ loga�a+ t ·2�) and decrease (pt =p/ loga�a + �T − t� · 2�) for low-fare classes, as weapproach departure.NLH. Nonhomogeneous Poisson with low-high

demand. We assume arrival rates decrease for high-fare classes (pt = p/ loga�a+ �T − t� · 2�) and increase(pt = p/ loga�a + t · 2�) for low-fare classes, as weapproach departure.

The log-factors are given by 2, and a is a constant,equal to the base of the logarithm (we take a= 2).

6.1.2. Running Time. Exact calculations of theoptimal expected revenue (DP), and expected valuesof the proposed policies (CEC, BPC) are practicallyimpossible. Already for two-leg networks (N2) withone class per itinerary, 50 seats per leg initial capac-ity and T = 300 time periods the computation takesabout two hours.

A tractable approach for measuring performance ofthe proposed policies, however, is provided by simu-lation. A simulation run of the CEC or BPC policiesfor a two-leg network takes less than a minute (lessthan a second per iteration). The largest network wesimulated was a four-leg network �N4�2� with twodemand classes per itinerary. When the initial capaci-ties are in the order of N = 50–100 and the time hori-zon has T = 300 time periods, a full simulation runfor such an instance takes a few minutes (i.e., a fewseconds per iteration).


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Table 2 Expected Value Calculation for �N2� with N= �50�50�; R= �25�20�35�; p= �04�03�01�; T = 10−200(HP) (NLH) (NHL)

T LP DP CEC BPC LP DP CEC BPC LP DP CEC BPC

10 195 195 195 195 2014 2014 2014 2014 2497 2497 2497 249720 390 390 390 390 4135 4135 4135 4135 4988 4988 4988 498830 585 585 585 585 6329 6329 6329 6329 7473 7473 7473 747340 780 780 780 780 8576 8576 8576 8576 9951 9951 9951 995150 975 975 975 975 1�0865 1�0865 1�0865 1�0865 1�2421 1�2421 1�2421 1�242160 1�170 1�170 1�170 1�170 1�3186 1�3186 1�3186 1�3186 1�4883 1�4881 1�4881 1�48870 1�365 1�365 1�365 1�365 1�5534 1�5522 1�5521 1�5522 1�7336 1�703 1�7028 1�690480 1�560 1�5595 1�5595 1�5595 1�7904 1�7553 1�7545 1�7549 1�8343 1�8009 1�7996 1�753490 1�755 1�7457 1�7454 1�7457 1�8853 1�8633 1�8608 1�8409 1�8461 1�8314 1�8295 1�7911100 1�950 1�8975 1�8964 1�8971 1�9251 1�9047 1�9014 1�8521 1�8584 1�8458 1�8431 1�8096110 2�020 1�9989 1�9966 1�9934 1�9373 1�922 1�917 1�8633 1�8711 1�8587 1�855 1�8225120 2�090 2�0646 2�0616 2�0316 1�9491 1�9343 1�9271 1�8769 1�8844 1�8721 1�8671 1�837130 2�140 2�1106 2�1073 2�0219 1�9606 1�9459 1�9367 1�8895 1�8983 1�8862 1�8798 1�8515140 2�170 2�1455 2�1405 2�0183 1�9717 1�9571 1�9462 1�8995 1�913 1�901 1�8932 1�864150 2�200 2�1757 2�1676 2�0189 1�9825 1�9681 1�9556 1�9072 1�9287 1�9167 1�9076 1�8758160 2�230 2�2024 2�1929 2�0193 1�9931 1�9787 1�9649 1�9131 1�9455 1�9336 1�9232 1�8878170 2�250 2�2228 2�2169 2�0194 2�0034 1�9891 1�974 1�9181 1�9638 1�952 1�9402 1�9008180 2�250 2�2362 2�233 2�0194 2�0135 1�9993 1�983 1�9229 1�9842 1�9725 1�9593 1�9155190 2�250 2�2438 2�2423 2�0194 2�0235 2�0093 1�9918 1�9286 2�0074 1�9958 1�9814 1�9324200 2�250 2�2475 2�2469 2�0194 2�0332 2�019 2�0005 1�9347 2�0353 2�0237 2�0081 1�9528

Note. The first section considers homogeneous arrivals, and the next two are nonhomogeneous arrivals with �= �50�50�30� and a= 2.

For more realistic airline networks [20 legs (10origins, 10 destinations, and one hub), 600 classes(100 itineraries, 6 fare classes)] the time per iterationis still in the order of a few minutes. This supportsthe idea that these dynamic policies can be used in anonline fashion to provide an adaptive control mech-anism where opportunity cost estimates are rapidlyrecalculated at each iteration.

6.1.3. Quality of Approximation

Expected Value Computations for Two-Leg Net-works. The results in Table 2 show expected valuecalculations for the case of the two-leg network,with different initial capacities, fare structures, anddemand regimes. In all cases, we observe that whenthe time horizon is large, the value of the CEC pol-icy is near optimal, whereas the bid-pricing policyappears to be off by a constant. When the time hori-zon is small, both policies perform optimally. Forintermediate stages when both some demand andcapacity constraints become binding, it is not clearwhich policy is better. Note that the BPC value is non-monotonic over time. We believe that this is because

of the dependence of the policy on the choice of dualsolutions (not enforced in our program).

Simulation Results. For two-leg networks we pro-vide in Table 3 a comparison between simulationresults and exact calculations for the DP value func-tion and expected values of the CEC and BPC poli-cies. The high dimensionality of the problem doesnot allow for exact expected value calculations forlarger networks. In Table 4, we present results from

Table 3 Simulations and Expected Value Calculation for Two-LegNetwork

CEC BPCDP

T LP EXP EXP SIM ( ) EXP SIM ( )

�N2�

50 975 975 975 973.3 (71) 975 973.2 (71)100 1�950 1�8975 1�8964 1,904.05 (83) 1�8974 1,903.75 (82)150 2�200 2�1755 2�1676 2,174.9 (45) 2�0188 2,021.2 (44)200 2�250 2�2475 2�2469 2,246.4 (18) 2�0194 2,023.3 (17)300 2�250 2�250 2�250 2,250 (0) 2�200 2,200 (0)

Note. �N2�: N= �50�50�; R= �25�20�35�; p= �04�03�01�.


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Table 4 Simulations for Larger Networks (Top-Down)

T LP CEC ( ) BPC ( )

�N3�

100 2,300 2,273 (140) 2,268 (140)200 3,200 3,155 (184) 3,148 (181)300 3,600 3,508 (178) 3,476 (177)

�N41�

100 3,500 3,462 (150) 3,460 (154)300 5,800 5,615 (78) 5,635 (74)600 6,050 5,951 (85) 5,842 (94)

�N42�

100 4,070 4,087 (213) 4,085 (216)200 6,125 5,914 (241) 5,856 (233)300 11,110 10,859 (302) 10,846 (309)

Note. �N3�: N = �60�60�100�; R = �10�20�50�30�and p = �02�03�04�; �N41�: Ni = 50, R = �20,20�50�30�60�40�45�55�, p = �02�025�005�005�01,015�01�01�; �N42�: Ni = 50, Rl = �20�20�50�30,60�40�45�55�, with Rh = 2 ·Rl and p = �015�002�02,001� 005� 0015� 005� 001� 01�001�015�0015�01,001�01�001�.

simulating various admission control policies forlarger networks. For simulation runs, we performNT = 100 trials and average the results to compute anexpected value estimate. We also compute the stan-dard deviation 3 of the estimation. In all cases, weobserve that our CEC policy performs better than theBPC policy, by up to 2%.

6.1.4. Robustness in Estimation. So far, we haveassumed that the CEC and BPC policies use cor-rect demand information, in the sense that the mean-demand forecast is unbiased. While this provides areasonable standard for comparing policies, in reality,the (expected demand) measurements obtained fromforecasting tools are seldom exact. We feel that it isof practical value to assess the robustness of the CECand BPC policies to noise and bias in the demandforecast.

To obtain a qualitative measure of how bias andnoise in the demand forecast affect these policies,we assess the impact on the underlying LP-value.Let � be an m-variate random variable represent-ing the noise in demand forecast for each class. Tomeasure the robustness of the LP-value to bias and

noise in demand forecasts, we compare LP�n�D� andE��LP�n�D+��. From the dual formulation we havethat

E��LP�n�D+��−LP�n�D�

= E��OC��n�D��≤ �R′ −v′ ·A�+ ·E��where v are the leg-shadow prices of LP�n�D�. Sothere is a sublinear effect on the value function, asso-ciated with the forecasting bias of the demand forthose classes whose fares exceed their bid prices.

Furthermore, computational experiments show thatthe CEC policy is surprisingly robust to certain formsof noise and bias in the demand data, much more sothan BPC.

Correlated Random Noise. At each point in timet we generate a multivariate random variable � ∼N�b��, to introduce noise into the arrival forecast.We propose two alternatives for adding noise to theforecast:

• Constant rate noise: Generate � and add it to thearrival rate at each time;

• Log-rate noise: Generate � and add �t = log�a+2 · �t−a�/T � ·� to the arrival rate.

From the results in Table 5, we observe that CECis constantly more robust than BPC to noise and bias.The columns LP� compute the LP-values with noisyforecasts: LP��n�Dt�= LP�n�Dt+E��t��.

Uncorrelated Random Noise. The goal of this com-putational exercise is to illustrate and compare therobustness of the BPC and CEC policies, to increasingbias in the demand forecast. We start with an initial(uncorrelated) bias per arrival b, and in each consecu-tive experiment subscripted �$b�, we increase the biasby a factor of $ = 1�2�4. In the last experiment (�),we introduce correlated noise (� same as above) inthe arrival rate estimation. We observe from Table 6that CEC is constantly more robust than BPC to noiseand bias.

6.1.5. Extensions

Rollout Heuristics. We examine how previouslycomputed values of the CEC and BPC policies can beused to provide estimates of opportunity costs, lead-ing to the roll-out heuristics R(CEC) and R(BPC). We


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Table 5 Computation with Noisy Forecasts

Constant Rate Log Rate

T LP LP� DP CEC BPC LP LP� DP CEC BPC

10 195 2137 195 195 195 195 2211 195 195 19520 390 4167 390 390 390 390 4087 390 390 39030 585 6119 585 585 585 585 6233 585 585 58540 780 7846 780 780 780 780 7655 780 780 78050 975 9751 975 975 975 975 9572 975 975 97560 1�170 1�1755 1�170 1�170 1�170 1�170 1�1235 1�170 1�170 1�17070 1�365 1�3664 1�365 1�365 1�365 1�365 1�3153 1�365 1�365 1�36580 1�560 1�6056 1�5595 1�5595 1�5595 1�560 1�569 1�5595 1�5595 1�559590 1�755 1�7703 1�7457 1�7452 1�7453 1�755 1�6855 1�7457 1�7452 1�7451100 1�950 1�9778 1�8975 1�8958 1�8954 1�950 1�8785 1�8975 1�8958 1�8946110 2�020 2�0861 1�9989 1�9961 1�9614 2�020 2�0085 1�9989 1�996 1�9747120 2�090 2�1152 2�0646 2�0617 1�9717 2�090 2�0692 2�0646 2�0616 1�9672130 2�140 2�1394 2�1106 2�1071 1�9801 2�140 2�1055 2�1106 2�1063 1�9812140 2�170 2�172 2�1455 2�1396 1�9877 2�170 2�1471 2�1455 2�1374 1�9879150 2�200 2�1978 2�1757 2�1669 1�9945 2�200 2�168 2�1757 2�1625 1�9935160 2�230 2�2278 2�2024 2�1928 1�9995 2�230 2�1788 2�2024 2�1855 2�0302170 2�250 2�250 2�2228 2�2168 2�0002 2�250 2�2032 2�2228 2�2084 2�1511180 2�250 2�250 2�2362 2�233 2�0002 2�250 2�2405 2�2362 2�2273 2�2089190 2�250 2�250 2�2438 2�2423 2�1306 2�250 2�2352 2�2438 2�2403 2�2234200 2�250 2�250 2�2475 2�2469 2�227 2�250 2�2485 2�2475 2�2468 2�2306

Note. �N2�: N= �50�50�; R= �25�20�35�; p= �04�03�01�; b= �007�−005�003�; � = 15

01 −002 004−002 008 001004 001 006

.

observe that the rollout procedure produces a signifi-cant improvement in the quality of the value functionapproximation, especially for the BPC policy (10%).This can be observed in Table 7 where we performedcalculations for various two-leg networks.

In practice, however, it is too expensive to com-pute and store all the H -values. In order to implementthis idea effectively, we suggest storing H -values fromseveral insightful a priori simulations, and interpo-lating these in an online fashion (as needed), for the“second time around” policy. It is an interesting ideato investigate what types of preprocessing simulationswould provide an insightful information database.

Simulations Using Monte Carlo Demand Estima-tion. We run a simplified version of the Monte Carlopolicy described in §3.3, that assigns the same weightsto all the trials and defines:

OCMCj �n� t�=

1r·r∑i=1

OCj �n� Dt−1i ��

When the number of trials r is large enough (so wehave a reasonably good MC-demand estimate), thispolicy provides a visible improvement (1�6%) overthe original version, as can be observed from the fol-lowing computational examples. However, when thenumber of trials is not large enough (30 or less),we have observed that this policy delivers a poorperformance. The tables below provide estimates forthe value of different models (LP, PILP, MC, CEC,BPC), as well as standard deviations, lower and upperbounds obtained through simulation. We also calcu-late the estimated average ratio P between differentmodels, as well as the standard deviation, minimum,and maximum ratio observed during simulation. Onecan observe for instance that the ratio CEC/BPC isalways ≥1, whereas the ratio MC/PILP is surprisinglyhigh, ranging between 96%–99%. The computationalresults presented in Table 8 show an improvement ofup to 20% of MC over CEC.


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Table 6 Increasing Bias in Arrival Rates

T DP LP CEC BPC LPb CECb BPCb LP2b CEC2b BPC2b

10 190 190 190 190 196227 190 190 202454 190 19020 380 380 380 380 3879062 380 380 3958123 380 38030 5680307 570 5679671 5680053 5789176 567958 5680149 5878351 5679583 568027140 7139429 740 7128882 7114275 7517866 7126471 7113734 7635732 712654 710612550 7848309 815 7827964 7450979 8189707 7821006 7433085 8229414 7816062 737354560 8198745 845 8179585 7496329 8491515 8175189 7480799 853303 8169229 74266470 8416218 855 8398851 7498441 855 8396463 7481471 855 8393091 74266480 8510686 855 8502119 7498441 855 8501653 7481471 855 8501115 74266490 8540811 855 8538794 7498441 855 8538777 7481471 855 8538765 74266499 8547912 855 8547523 7498441 855 8547523 7481471 855 8547523 742664

T DP LP4b CEC4b BPC4b LP� CEC� BPC�

10 190 2149079 190 190 1918341 190 19020 380 4116247 380 380 361348 380 38030 5680307 6056702 5679585 5680273 6714161 5679394 567818840 7139429 7871465 7125613 7061965 7074845 7121564 708948850 7848309 8308828 7800541 723385 8011911 7797808 75261860 8198745 855 8150074 728399 8319429 8154683 757347270 8416218 855 8385854 728399 855 8385076 757558980 8510686 855 8500349 728399 855 8499272 75780490 8540811 855 8538752 728399 855 8538605 76114299 8547912 855 8547523 728399 855 8547513 7638937

Note. �N2�: Data: N= �19�19�; R= �25�20�35�; p= �03�04�01�, b= �007�−005�003�.

6.2. Computational Performance UnderCancellations and Overbooking

The objective in this section is to understand the rela-tive performance of BPC and CEC in an environmentwith cancellations and overbooking. We considered abooking horizon of 15 periods for a hub and spokenetwork with five cities and two classes, of the type�N4�2�. The arrival process for the highest fare classis nonhomogeneous Poisson with rate 0.5 for Peri-ods 1–13 and 5 for Periods 14, 15. The arrival pro-cess for the lowest fare class is homogeneous Poissonwith Rate 3. For simplicity, we kept the fare of thehigher class in a single-leg itinerary equal to $100 andof the lower class equal to $80. We varied the fare oftwo-leg itineraries. After experimentation we identi-fied the following parameters that affected the relativeperformance of CEC and BPC: We varied the follow-ing parameters: (a) The overbooking penalty (Cj ), (b)the probability of cancellation (pc), and (c) the fare ofa two-leg itinerary compared to a single-leg itinerary.

The implementation was done in C and the algo-rithms were run on a Dell Pentium III 600MHz oper-ating under Linux.

In Table 9, we report the behavior of CEC andBPC as a function of the overbooking penalty. Weobserve that CEC leads to consistently higher revenueby approximately 1%.

In Table 10, we report the behavior of CEC and BPCas a function of the cancellation probability. With theexception of very high cancellation rate (0.30), CECoutperforms BPC.

In Table 11, we report the behavior of CEC and BPCas a function of the ratio 2 of the fare of a two-legitinerary versus the fare of a single-leg itinerary. CECoutperforms BPC, but the level of overperformancedecreases as 2 ranges from 1 to 2.

7. ConclusionsIn this paper, we have presented several models andalgorithms for solving the stochastic and dynamic


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Table 7 Rollout Heuristics

T LP DP CEC BPC R(CEC) R(BPC)

�N21�

1 195 195 195 195 195 19510 195 195 195 195 195 19520 390 390 390 390 390 39030 585 585 585 585 585 58540 780 780 780 780 780 78050 975 975 975 975 975 97560 1�170 1�170 1�170 1�170 1�170 1�17070 1�365 1�365 1�365 1�365 1�365 1�36580 1�560 1�5595 1�5595 1�5595 1�5595 1�559590 1�755 1�7457 1�7454 1�7457 1�7456 1�7457100 1�950 1�8975 1�8964 1�8971 1�8972 1�8974110 2�020 1�9989 1�9966 1�9934 1�998 1�9955120 2�090 2�0646 2�0616 2�0316 2�0632 2�0422130 2�140 2�1106 2�1073 2�0219 2�1089 2�0528140 2�170 2�1455 2�1405 2�0183 2�143 2�1042150 2�200 2�1757 2�1676 2�0189 2�1729 2�1639160 2�230 2�2024 2�1929 2�0193 2�1996 2�1972170 2�250 2�2228 2�2169 2�0194 2�2206 2�2174180 2�250 2�2362 2�233 2�0194 2�2348 2�2302190 2�250 2�2438 2�2423 2�0194 2�2431 2�2381200 2�250 2�2475 2�2469 2�0194 2�2472 2�2419

�N22�

1 19 19 19 19 19 1910 190 190 190 190 190 19020 380 380 380 380 380 38030 570 5680307 5679671 5680053 5680285 568025840 740 7139429 7128882 7114275 7138018 713251750 815 7848309 7827964 7450979 7843939 774430160 845 8198745 8179585 7496329 8192368 815858170 855 8416218 8398851 7498441 8406865 83707280 855 8510686 8502119 7498441 8506009 846444290 855 8540811 8538794 7498441 8539772 8487245100 855 8548245 8547925 7498441 8548099 8498084

Note. �N21�: N = �50�50�, R = �25�20�35�, p = �04�03�01�; �N22�: N = �19�19�, R =�25�20�35�, p= �03�04�01�.

NRM problem. We proposed a new efficient algo-rithm, based on a certainty equivalent approximationand compared it with the widely used bid-price con-trol policy. This policy conceptually improves the cur-rent NRM-approach based on additive bid pricing,by using more insightful, piecewise linear approxima-tions of opportunity cost. It is just as easy to compute,but as opposed to BPC, it is more “robust” in the fol-lowing sense:

• The CEC opportunity cost estimates are uniquelydefined at each state, whereas for additive bid prices,there may be several dual optimal solutions.

• The CEC policy is optimal in the deterministicregime, whereas BPC is not.

• Computationally we observe that the CEC pol-icy outperforms the BPC policy when the load factors(the ratio of expected demand to available capacity)tend to be large. When the load factor is small, bothpolicies perform optimally. There is a critical range


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Table 8

�N21� LP PILP MC CEC BPC P AvgP StdP MinP MaxPEXP 2�200 2�1912 2�170 2�1628 2�1616 CEC/BPC 10006 00035 09931 10074Std 0 4448 5045 5241 5419 CEC/PILP 0987 00093 09569 1LB 2�200 2�070 2�035 2�000 2�000 MC/PILP 09903 00084 09645 1UB 2�200 2�250 2�250 2�250 2�240 MC/CEC 1003 00067 09909 10225

�N22� LP PILP MC CEC BPC P AvgP StdP MinP MaxPEXP 225 215 208 205 2025 CEC/BPC 101 0028 1 1075Std 0 1055 1653 187 1976 CEC/PILP 09521 00498 08537 1LB 225 195 185 175 175 MC/PILP 09664 0038 09024 1UB 225 225 225 225 225 MC/CEC 1016 0034 1 10857

�N23� LP PILP MC CEC BPC P AvgP StdP MinP MaxPEXP 8�000 7�922 7�785 7�771 7�766 CEC/BPC 10007 0011 0975 102Std 0 2243 2387 2852 2838 CEC/PILP 09808 00166 094 1LB 8�000 7�500 7�200 7�050 7�050 MC/PILP 09828 0017 0939 1UB 8�000 8�600 8�400 8�500 8�400 MC/CEC 1002 00184 09615 10425

�N4� LP PILP MC CEC BPC P AvgP StdP MinP MaxPEXP 6�800 n.a. 6�6944 6�588 6�5688 CEC/BPC 10029 00051 09935 10117Std 0 n.a. 21444 21373 215 MC/CEC 101624 00127 09908 10421LB 6�800 n.a. 6�260 6�140 6�160UB 6�800 n.a. 7�065 6�950 6�930

Note. �N2�: N= �50�50�, R= �25�20�35�, p= �03�04�01�, T = 150, NT = 100, r = 50; �N21�: N= �5�5�, R= �25�20�35�, p= �03�04�01�, T = 20, NT =10, r = 50; �N23�: N= �20�20�, R= �25�20�35�, p= �02�03�05�, T = 50, NT = 50, r = 50; �N4�: N= �50�50�100�100�, R= �20�20�50�30�60�40�45�55�,p= �015�015�015�015�01�01�01�01�, T = 200, NT = 50, r = 60.

when the load factor is close to one, where it is notclear which policy provides a better performance, butthe difference between the two is very small. More-over, computational exercises show that the value ofthe CEC policy is very close to the value function DP,and to the perfect information upper bound PILP.

• The CEC outperforms the BPC policy in anenvironment where cancellations and overbookingare present under various scenarios of cancella-tion probabilities, disparity in fares and overbookingpenalties.

Table 9 The Expected Revenue in 200 Simulation Runs as aFunction of the Overbooking Penalty

Cj 100 110 120 130

CEC 24,480 24,450 23,975 23,890BPC 22,355 22,270 21,920 21,567

Note. The cancellation probability was 0.01 and the fare of a two-legitinerary was equal to the fare of a single-leg itinerary.

• The CEC policy is significantly more robust thanBPC to bias and (correlated) noise in the demandforecast.

The following extensions provide insights towardsfurther developments:

—The rollout procedure produces an importantimprovement in the value of both policies, virtuallyclosing the optimality gap.

—The Monte Carlo simulation procedure incor-porates demand variability, and produces a significant improvement in the value of the CEC policywhen the number of trials is sufficiently large.

Table 10 The Expected Revenue in 200 Simulation Runs as aFunction of the Overbooking Penalty

pc 0.05 0.10 0.20 0.30

CEC 22,382 21,095 18,760 11,280BPC 19,680 20,160 17,262 12,960

Note. The overbooking penalty was $130 and the fare of a two-legitinerary was equal to the fare of a single-leg itinerary.


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Table 11 The Expected Revenue in 200 SimulationRuns as a Function of the Ratio � of theFare of a Two-Leg Itinerary Versus theFare of a Single-Leg Itinerary

� 1 1.5 2

CEC 22,522 23,400 24,457BPC 20,587 21,437 24,187

Note. The cancellation probability was 0.1 and the over-booking penalty $130.

AcknowledgmentsThe authors thank Eduardo Messmacher for implementing thealgorithms with cancellation and overbooking. They are also grate-ful to two anonymous referees for many useful suggestions.

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Received: October 2000; revision received: September 2001; accepted: January 2002.


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