An effective dynamic decision policy for the revenue management of an airline flight

Int. J. Production Economics 144 (2013) 440–450

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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe

An effective dynamic decision policy for the revenue management of an airline flight Yi-Feng Hung n, Chien-Hao Chen Department of Industrial Engineering and Engineering Management National Tsing Hua University, 101 Kuang Fu Road, Section 2, Hsinchu 30013, Taiwan, ROC

art ic l e i nf o

a b s t r a c t

Article history: Received 9 March 2012 Accepted 1 March 2013 Available online 21 March 2013

Airline companies normally classify the seats of a cabin class into a number of fare classes. Customers of different fare classes arrive randomly during the booking horizon. Every time a customer in a certain fare class arrives, the airline company must decide promptly whether to fulfill or reject the request. To increase revenues, the airline company may reject certain lower-fare class customer requests and reserve the seats for future higher-fare class customers. However, rejecting too many lower-fare class customers may result in empty seats when the flight takes off. Given the multiple fare classes of a flight and the non-homogeneous Poisson customer arrival process in each fare class, and with the aim of maximizing the revenue of the flight, this study develops and tests two heuristic approaches – the dynamic seat rationing (DSR) decision policy and the expected revenue gap (ERG) decision policy – to help the airline make a fulfillment-or-rejection decision when a customer arrives. The simulation experiments show that ERG performs best among all tested approaches and, on average, the revenue from the ERG being merely 0.8% less than that of the optimal decision made with perfect information. Moreover, the ERG is very robust under various problem conditions. & 2013 Elsevier B.V. All rights reserved.

Keywords: Revenue management Rationing decision Decision support

1. Introduction Optimally allocating finite and perishable inventory to different price classes is a critical issue for a profit-oriented organization. It is obvious that selling inventory items to more profitable orders will increase the revenue, especially when the expected demand is higher than the available inventory. This order fulfillment (or order selection) decision is the focus of revenue management, which has been developed for a number of different industries in the last few decades. For example, recently, Hung and Lee (2010) worked on manufacturing capacity rationing problem, while Hung et al. (2012), Pinto (2012), and Hung and Hsiao (2013) investigated inventory rationing problem. This study focuses on revenue management for the airline industry. After the deregulation of the airline industry in the 1970s, airline companies were allowed to sell seats in the same cabin for different prices. The reason behind revenue management is that different customers may have different acceptable prices (valuations); thus, airlines will try to sell at a higher price to those customers who are willing to pay a higher price and a lower price to those who will only accept a lower price. Customers arrive randomly during the booking horizon of a flight, which is the

n

Corresponding author. Tel.: þ886 3 5742939; fax: þ 886 3 5722685. E-mail address: [email protected] (Y.-F. Hung).

0925-5273/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpe.2013.03.012

duration between the start time for making reservations and the time when the flight takes off. Since the demand for the highest fare class is not enough to fill up a flight, airlines cannot keep all the seats for the highest fare class customers. Also, since the customers of different fare classes arrive at different times during the booking horizon, an airline should not sell too many seats to lower-fare class customers during the early stage of the booking horizon. On the other hand, at the end of the booking horizon, if there are remaining seats, the airline loses revenue due to perished inventory—empty seats. The airline wants not only to sell more tickets to higher-fare class customers but also to sell all of the available seats. Solving such a stochastic trade-off decision problem is the focus of this study. Airlines classify the seats in a cabin class into different fare classes. The normal practice is that the airline allocates available seats into these fare classes based on the demand forecast for customers with different acceptable prices, and they will reserve the allocated seats for future higher-fare class customers. The number of reserved seats is called the protection level, and the booking limit is calculated by subtracting the protection level from the total available seats. This seat allocation problem has been investigated in full swing, and various approaches have been proposed since the deregulation of the airline industry. In addition, the solution methods can be divided into static and dynamic ones (Lautenbacher and Stidham, 1999; Pak and Piersma, 2002). The static methods assume that the lower-fare class customers always

Y.-F. Hung, C.-H. Chen / Int. J. Production Economics 144 (2013) 440–450

come before the higher-fare class customers, so the seat allocation problem is computed at the beginning of the booking horizon (Belobaba, 1987; Brumelle and McGill, 1993; Littlewood, 1972). On the other hand, dynamic methods, without assuming arrival sequence, can adjust the booking limits dynamically based on the latest status when a customer arrives (Lautenbacher and Stidham, 1999; Lee and Hersh, 1993). The methods proposed in this study are dynamic ones. Pak and Piersma (2002) provided an overview of operations research techniques in solving the seat allocation problem. They classified the methods into single-leg models and network models. In a single-leg model, each flight leg is optimized independently, while in a network model, the objective is to maximize the total revenue of multiple legs. Littlewood (1972) first proposed a static method to solve the single-leg seat allocation problem with two fare classes and used marginal revenues to calculate the number of seats reserved for high-fare class customers. Extending from Littlewood's method, Belobaba (1987) proposed an expected marginal seat revenue method to solve the problem of more than two fare classes. In addition to Belobaba's work, Curry (1990) developed methods to find the optimal booking limits for multiple fare classes. Also, Brumelle and McGill (1993) introduced an approach to solve the optimal protection levels and showed that the obtained revenue is greater than the one by Belobaba. Lee and Hersh (1993) divided the booking horizon into several small decision periods and assumed that at most only one customer arrives in a period. They then approximated the problem using a discrete-time dynamic programming model, which employs a recursive function to calculate marginal revenues and determine whether or not an arrival customer should be accepted. Subramanian et al. (1999) added the assumptions of overbooking, cancellations, and no-shows into the model proposed by Lee and Hersh. Liang (1999) reformulated the assumptions of discrete decision periods of Lee and Hersh's model into a continuous-time frame. Lautenbacher and Stidham (1999) proposed two Markov decision process models for static and dynamic methods and constructed an omnibus model that assumed there could be more than one customer in a small decision period. Not using demand forecast, Van Ryzin and McGill (2000) introduced an adaptive approach to adjust the protection levels by historical observations. You (2001) developed a dynamic programming model taking into account that rejected lower-fare class customers may be upgraded to the higherfare class. Haerian et al. (2006) divided the previous booking policies into two booking policies. One is right booking policy, (also called standard nesting policy) and the other is left booking policy (also called theft nesting policy). Glover et al. (1982) were the first to describe the multiple-leg seat allocation problem as a network flow problem and proposed a linear programming model. This model considered a deterministic demand and reserved the exact expected demand seats for the higher-fare class. Williamson (1992) developed an expected marginal revenue model with a probabilistic demand, but there were a large number of decision variables. Extending from Williamson's model, de Boer et al. (2002) proposed a stochastic linear programming model which has fewer decision variables. By using an idea similar to that of dynamic programming, Bertsimas and de Boer (2005) introduced a simulation-based optimization algorithm to improve the booking limits. The focus of this study is to develop effective dynamic policies that can be applied when a customer arrives to solve the stochastic decision problem with the objective of maximizing the revenue of a flight and without using booking limits.

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1. More than one fare class: Since the prices customers are willing to pay are not the same and the price can be different for the same cabin class, the seats are classified into a discrete number of fare classes. 2. Non-homogeneous Poisson arrival processes: During the booking horizon of a flight, customers in different fare classes arrive randomly. From past experience, the customers searching for a low price (lower-fare class) usually arrive in the early stage of a booking horizon. These customers are price-sensitive and, normally, are tourists who have a well-prepared plan in advance. On the other hand, the customers who can afford to pay a high price (higher-fare class) often come in the later stage; usually, they are business travelers. Most of the previous studies assume that the customer arrival process is homogeneous Poisson; that is, the arrival rate is constant during the booking horizon (Haerian et al., 2006; Lee and Hersh, 1993; Liang, 1999). However, usually, the expected arrival rate fluctuates during a booking horizon. This study assumes that the arrival rate is a function of time and fare class. 3. Independent processes: The customer arrival process of a fare class is independent of the arrival process of another fare class. 4. No overbooking, no cancellations, and no no-shows: Customers whose requests are fulfilled will not cancel their reservations and will show up at the time when the flight takes off. Therefore, the airlines will not book more customers than there are available seats. 5. No demand recapturing: A customer strongly specifies a fare class, and this fare class cannot be upgraded to a higher fare class. If the specified fare class is full, the rejected customer will be lost forever. 6. One seat request: A customer requests only one seat. A customer requesting more than one seat is modeled as several individual arrivals in the non-homogeneous Poisson process. 7. Fixed booking horizon: The duration of the booking horizon for a flight is a known constant. Each time a customer arrives, by weighting the chance of future higher-fare class customers and chance of empty seat at take-off time, the airline should promptly make a fulfillment-or-rejection decision with the objective of maximizing the total revenue generated by the flight. Under the above assumptions, this study proposes two new dynamic decision policies to handle the decision problem and compares the proposed approaches with several existing ones which are applicable under the assumptions made here.

3. Decision policies This section presents two decision policies: dynamic seat rationing (DSR) and expected revenue gap (ERG). Section 3.1 summarizes the known parameters for the two proposed decision policies. Sections 3.2 and 3.3 thoroughly discuss the two policies, respectively. By using a small example, Section 3.4 demonstrates the two proposed approaches. 3.1. Known parameters for decision policies Let j be the index for fare class (or, briefly, class) and j¼1,2,3,…,J. The smaller the index value, the higher the fare class. J is the number of fare classes considered. The following parameters are assumed to be known at decision making time:

2. Concepts and assumptions This section describes the problem and the assumptions of the considered revenue management problem for a flight in this study.

s0 h λj(τ)

the initial total available seats at time 0. the duration of the booking horizon. the expected arrival rate of class j customers at time τ.

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fj t

the price of fare class j. the current time, or when a customer arrives. The airline should make a fulfillment-or-rejection decision at this time. the requested fare class of the current arrival customer. the number of current available seats, which is equal to s0 subtracted by the number of the total fulfilled customers within the time interval (0, t).

k st

the correct decision. Thus, pr is the probability that rejecting the current request is the correct decision. Let Ef be the event in which, after fulfilling the current request, the total future customer requests with classes higher than k are less than or equal to the remaining seats, s−1. We have the probability of Ef, pf ¼ PðEf Þ ¼ Pð0≤N kt ≤s−1Þ. Since, under the event Ef, we should fulfill the current request; that means fulfilling the current request is the correct decision. Thus, pf is the probability that fulfilling the current request is the correct decision. e−nkt ðnkt Þx : x! x¼s

∞

∞

3.2. Optimal decision under perfect information

pr ¼ Pðs≤Nkt Þ ¼ ∑ PðNkt ¼ xÞ ¼ ∑

An optimal decision method called decision under perfect information (DUPI) is described in this subsection. First, we assume that the complete information on the classes and the arrival times of all customers are known at the beginning of the booking horizon. Let zj be the actual number of class j customers who arrived during the booking horizon, and Aj be the integer variable of the number of seats allocated to class j customers. The following integer programming is formulated to obtain an optimal solution:

Note that PðN kt ¼ xÞ ¼ e−nkt ðnkt Þx =x! approaches to zero, as x approaches to ∞. For computation purposes, a sufficiently large x will make the Poisson probability PðNkt ¼ xÞ less than ε, where ε is a very small value, and thus, we treat such a probability PðNkt ¼ xÞ as zero. Therefore, we only need to compute a finite number of ∞ −nkt terms in the summation of ∑∞ ðnkt Þx =x!. x ¼ s PðN kt ¼ xÞ ¼ ∑x ¼ s e −nkt Similarly, pf ¼ Pð0≤Nkt ≤s−1Þ ¼ ∑s−1 PðN ¼ xÞ ¼ ∑s−1 kt x¼0 x ¼ 0e x ðnkt Þ =x!, where s−1 is a limited number. Since the two decisions are mutually exclusive, the request should be either rejected or fulfilled. The airline will make a fulfillment-or-rejection decision based on which decision has a higher probability of being correct. That is, when a customer arrives, if the computed pf is greater than or equal to the computed pr, the request should be fulfilled. On the other hand, if pr is greater than pf, the request should be rejected.

J

Maximize ∑ f j Aj , j¼1

subject to J

∑ Aj ≤s,

x¼s

j¼1

Aj ≤zj ,j ¼ 1,:::,J: By observing the formulation, an optimal decision can also be generated by a simple greedy rule that starts fulfilling from the highest class customers to the lowest class customers until the exhaustion of the available seats. Let aj be the optimal value of Aj. Following the greedy rule, aj ¼ MINfMAXðs−∑j−1 j′ ¼ 1 aj′ ,0Þ,zj g, j¼1,...,J. However, in an actual environment, perfect information does not exist and DUPI approaches cannot be applied. 3.3. Dynamic seat rationing decision policy

The second heuristic approach, called expected revenue gap (ERG) decision policy, is discussed in this subsection. 3.4.1. Approximation of the expected revenue under DUPI The numbers of future arrival customers of various classes are Poisson random variables. By knowing the distributions of these Poisson random variables, we can generate a scenario tree that represents all possible scenarios. All of the possible arrival numbers of class j customers are branched from a level (j−1) node. That is, a level j node represents a possible number of class j arrivals. A scenario tree is shown in Fig. 1. In fact, the scenario tree is infinite because the number of possible values of a Poisson random variable is infinite. However, the probability of a sufficiently large number of arrivals or sufficiently small number of arrivals will be too small for computation purposes and can be omitted from the tree without the loss of computation accuracy. For class j, the probability of x arrivals PðN jt ¼ xÞ ¼ e−njt ðnjt Þx =x!, where x¼0,1,2,.... Note that if x is sufficiently large or if njt is sufficiently large and x is sufficiently small, the Level 0

(Class 1) Level 1

(Class 2) Level 2

...

1

0 1 2

...

...

...

...

...

Fig. 1. A scenario tree for ERG.

...

...

...

1

...

...

...

...

... ... ...

...

... ... ...

...

2

0 1 2

...

...

... 1

0

...

0

0 1

(Class J) Level J

...

2

(Class J-1) Level J-1

...

1

...

0

...

0

...

The first heuristic approach for the revenue management of a flight proposed by this study is called dynamic seat rationing (DSR) decision policy that is modified from the dynamic stochastic capacity rationing (DSCR) procedure proposed by Hung and Lee (2010) to perform revenue management for make-to-order manufacturing. It is discussed as follows. At the time of the decision-making process (when a customer arrives), the parameters t (the current time), k (the fare class of the current arrival customer), and s (the current available seats) are known. This study assumes that the customer arrival of class j is a nonhomogeneous Poisson process with expected arrival rate λj(τ) at time τ. According to the non-homogeneous Poisson process properties provided by Ross (2006), the number of class j customers that arrive within time interval (t, h] is a Poisson random Rh variable Njt with mean njt ¼ EðN jt Þ ¼ t λj ðτÞdτ. By the additive property of Poisson distribution, the total number of customers with classes higher than k that arrive within time interval (t, h] is also a Poisson random variable Nkt with mean nkt ¼ EðNkt Þ ¼ k−1 k−1 Eð∑k−1 j ¼ 1 N jt Þ ¼ ∑j ¼ 1 EðN jt Þ ¼ ∑j ¼ 1 njt . Let Er be the event in which the total future customer requests with classes higher than k are greater than or equal to the current available seats s. We have the probability of Er, pr ¼ PðEr Þ ¼ Pðs≤N kt Þ. Since, under the event Er, we should not fulfill the current request; this means that rejecting the current request is

3.4. Expected revenue gap decision policy

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probability P(Njt ¼x) is very close to zero and can be approximated as zero. Therefore, we only need to consider a finite number of x values. Let Uj be the set of the numbers of class j arrivals whose probabilities are significant; that is, Uj ¼{x|P(Njt ¼ x)≥ε}, where ε is a very small value. Thus, the number of nodes branching from a level (j−1) node is n(Uj), the number of elements in Uj. A leaf node of the tree represents a scenario in which the number of arrivals of each class is known. Therefore, under a scenario, the optimal decision under perfect information (DUPI) can be obtained by the greedy rule. Let y be the index for scenarios. Let zyj be the number of class j customers in scenario y, and ayj be the optimal number of seats allocated to class j customers by DUPI under scenario y. Thus, given s seats, the revenue of the DUPI in scenario y is Ry ¼ ∑Jj ¼ 1 f j ayj . In addition, the probability of scenario y is given by py ¼ ∏Jj ¼ 1 PðN jt ¼ zyj Þ. Then, the expected revenue of DUPI is calculated by EðRt ðsÞjD ¼ DUPIÞ ¼ ∑y py Ry ¼ ∑y ð∏Jj ¼ 1 PðN jt ¼ zjy ÞÞð∑Jj ¼ 1 f j ajy Þ. 3.4.2. Decision tree The number of arrival customers in each class between present time and take-off time of a flight is unknown at the present time and, thus, is a random variable. Further, the revenue generated from selling current available seats to future arrival customers is also unknown and is a random variable. Each time a customer arrives, a fulfillmentor-rejection decision should be made by a certain decision model, which will certainly affect the revenue. Thus, given decision policy D, the future revenue by selling s seats from time t to the take-off time is a random variable Rt(s)|D. It depends not only on the current time t and the current available seats s, but also on the decision policy D. Assume that a class k customer arrives at time t and is willing to pay price fk to purchase one seat. The revenues under the two decisions – fulfill and reject – can be illustrated by the decision tree in Fig. 2. In the figure, t− denotes the time that is infinitesimal time earlier than time t. When the customer arrives, if the airline fulfills the customer request, the number of remaining seats decreases by one and the revenue of the flight increases by the price of the class the customer requests; that is, the revenue by fulfilling the current request is Rt(s−1)|Dþfk. On the other hand, if the airline rejects the request, both the number of available seats and the revenue remain the same. That is, if the current request is rejected, the revenue is Rt(s)|D. However, we are not able to draw the decision tree further down, and we do not know how many levels are needed before it reaches a leaf node since the next arrival time is a random variable following exponential distribution and the number of arrivals is a random variable following Poisson distribution. Since both Rt(s)|D and Rt(s−1)|D are random variables, a rational decision maker will choose fulfillment or rejection by comparing the expected values of Rt(s−1)|Dþfk and Rt(s)|D. That is, if E(Rt(s)|D) 4E(Rt(s−1)|D) þfk, the decision should be to reject the request. If E (Rt(s)|D)≤E(Rt(s−1)|D) þfk, the decision should be to fulfill the request. However, it is difficult to analytically describe the random variable Rt(s)|D and Rt(s−1)|D; thus, their close forms and expected values are unknown. Define random variable Gt(s)|D ¼Rt(s)|D−Rt(s −1)|D. Thus, since E(Gt(s)|D) ¼E(Rt(s)|D)−E(Rt(s−1)|D), the condition for rejection and fulfillment decisions becomes E(Gt(s)|D)4 fk and E (Gt(s)|D)≤fk, respectively. The E(Gt(s)|D) here can be treated as the opportunity cost of utilizing decision policy D to sell s seats during

Fig. 2. The revenues under the two different decisions.

443

the time interval (t,h). Thus, the decision of fulfillment-or-rejection should be based on comparing the opportunity cost with the acceptable price of the current arrival customer. 3.4.3. The ERG policy E(Gt(s)|D ¼ERG) denotes the expected revenue gap under the decision policy ERG. As discussed previously, each time a customer arrives, the airline can make a decision by comparing the opportunity cost E(Gt(s)|D¼ERG) with fk. If E(Gt(s)|D ¼ERG)≤fk, the request should be fulfilled. Otherwise, if E(Gt(s)|D ¼ERG)4fk, the request should be rejected. However, it is difficult to analytically describe the random variables Rt(s)|D and Gt(s)|D for any decision policy D. For decision making purposes, we only need to know their expected values. In the proposed ERG policy, the E(Gt(s)|D¼ ERG) is approximated by E(Gt(s)| D¼DUPI); that is, E(Gt(s)|D¼ ERG)≈E(Gt(s)|D¼ DUPI)¼E(Rt(s)|D¼ DUPI) −E(Rt(s−1)|D¼ DUPI). The computation of the approximated values for E(Rt(s)|D¼DUPI) and E(Rt(s−1)|D¼ DUPI) is discussed in Section 3.4.1. We conjecture that the expected value E(Rt(s)|D¼ ERG) could be very close to E(Rt(s)|D¼DUPI). Furthermore, since (Gt(s)|D)¼(Rt(s)|D)−(Rt(s −1)|D), we conjecture that the difference between E(Gt(s)|D¼ ERG) and E(Gt(s)|D¼ DUPI) may be even smaller than the difference between E (Rt(s)|D¼ ERG) and E(Rt(s)|D¼DUPI). To validate this conjecture, a simulation experiment is performed in Section 4.4. 3.5. A numerical example A simple example is used to demonstrate the two proposed decision policies, DSR and ERG. We consider a problem with three classes, and the prices for class 1, class 2, and class 3 are 300, 200, and 100, respectively. There are 10 initial available seats. The duration of the booking horizon is 10 days (h) and the booking horizon is divided into 2 intervals; thus, the length of each interval is 5 days. Each arrival customer requests one seat. The expected arrival rates are defined as follows: ( ( 0:2 if τ∈½0,5Þ 0:6 if τ∈½0,5Þ λ1 ðτÞ ¼ ; λ2 ðτÞ ¼ ; 0:6 if τ∈½5,10 0:6 if τ∈½5,10 ( 1:1 if τ∈½0,5Þ λ3 ðτÞ ¼ : 0:5 if τ∈½5,10 The arrival numbers, classes, prices, and arrival times of all customers during the booking horizon are shown in the first four columns of Table 1. Table 1 The computation results of FCFS, DSR, ERG, and DUPI for the example. Customer arrival #

Class Price Arrival time

FCFS

DSR decisions

ERG decisions

DUPI

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

3 3 2 1 2 3 3 2 3 3 3 2 1 2 3 2 1 1

Accept Accept Accept Accept Accept Accept Accept Accept Accept Accept Reject Reject Reject Reject Reject Reject Reject Reject

Reject Accept Accept Accept Accept Reject Reject Accept Reject Reject Reject Accept Accept Accept Reject Accept Accept Reject

Reject Reject Accept Accept Accept Reject Reject Accept Reject Reject Reject Accept Accept Accept Reject Accept Accept Accept

Reject Reject Accept Accept Accept Reject Reject Accept Reject Reject Reject Accept Accept Accept Reject Accept Accept Accept

100 100 200 300 200 100 100 200 100 100 100 200 300 200 100 200 300 300

0.3245 0.9564 1.3758 2.1124 2.9578 3.3145 3.8547 4.0421 4.4703 4.9512 5.2152 5.8545 6.5258 7.064 7.2456 8.2977 8.9221 9.2514

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3.5.1. FCFS in the example The first approach is to use a first-come-first-served (FCFS) rule, in which every arrival customer request is fulfilled as long as there is a remaining seat. Thus, in this example, the FCFS rule fulfills the first 10 customers, and the revenue is $1500. 3.5.2. DSR in the example The second approach uses the DSR decision policy that compares the probability that fulfilling the request is a correct decision with the probability that rejecting the request is a correct decision. 3.5.2.1. At time t¼ 0.3245. The first customer arrives. Since the number of available seats s is greater than zero and the customer does not belong to class 1, the procedure should continue without knowing the information of customers arriving afterwards. Because the customer belongs to class 3, the number of customers with classes higher than 3 within time interval (0.3245, 10] is a Poisson random variable with mean Z 10 Z 10 nkt ¼ n1t þ n2t ¼ λ1 ðτÞdτ þ λ2 ðτÞdτ u ¼ 0:3245

u ¼ 0:3245

¼ ½0:2  ð5−0:3245Þ þ 0:6  5 þ ½0:6  ð5−0:3245Þ þ 0:6  5 ¼ 9:7404: Since there are 10 available seats, the probability that rejecting the request is a correct decision, pr ¼ Pð10≤N kt Þ ¼

∞

∑ PðN kt ¼ xÞ ¼

x ¼ 10

e−9:7404 ð9:7404Þx : x! x ¼ 10 ∞

∑

At the current time, if x is greater than or equal to 22, PðN kt ¼ xÞ is less than 10−6 (ε value). Thus, pr ¼ ∑21 x ¼ 10 PðN kt ¼ xÞ ¼ 0:5092 If the airline decides to fulfill the current request, the number of remaining seats is 9 (¼10−1). Similarly, the probability that fulfilling the request is a correct decision, 9

e−9:7404 ð9:7404Þx ¼ 0:4908 x! x¼0 9

pf ¼ Pð0≤N kt ≤9Þ ¼ ∑ PðN kt ¼ xÞ ¼ ∑ x¼0

Since pr is greater than pf, the airline should reject the current request and the number of seats remains 10. 3.5.2.2. When time t40.3245. Each time a customer arrives, the same computation procedure can be performed. The DSR computation results for the rest of the arrivals are shown in Table 1 and its total revenue is $2200. 3.5.3. ERG in the example The third approach is to use the ERG policy that compares the expected revenue gap under DUPI with the price of the current arrival customer. 3.5.3.1. At time t¼0.3245. The first customer arrives and the information of the customers arrive afterwards is unknown. The numbers of arrival customers for class 1, class 2, and class 3 are Poisson random variables, N1t, N2t and N3t, with mean n1t ¼ 0:2  ð5−0:3245Þ þ 0:6  5 ¼ 3:9351, n2t ¼ 0:6  ð5−0:3245Þ þ 0:6  5 ¼ 5:8053, n3t ¼ 1:1  ð5−0:3245Þ þ 0:5  5 ¼ 7:643, respectively.

Table 2 The gap below DUPI of the three approaches. Approach

FCFS

DSR

ERG

Gap below DUPI

37.5%

8.33%

0%

For computation purposes, the probability P(Njt ¼ x) that is less than 10−6 is approximated as zero. Therefore,     e−3:9351 ð3:9351Þx 4 10−6 ¼ f0,1,2,:::,16g U 1 ¼ xPðN1t ¼ xÞ ¼ x!     e−5:8053 ð5:8053Þx 4 10−6 ¼ f0,1,2,:::,20g U 2 ¼ xPðN2t ¼ xÞ ¼ x!     e−7:643 ð7:643Þx 410−6 ¼ f1,2,3,:::,24g: U 3 ¼ xPðN3t ¼ xÞ ¼ x! The scenario tree for the first arrival customer can be generated. Based on the scenario tree, we can calculate the revenue of DUPI and the probability for each scenario. For example, in a scenario (say, scenario y′), the numbers of arrivals for class 1, class 2, and class 3 are 3, 5, and 6, respectively. Since the number of available seats is 10, the optimal number of seats allocated to the three classes are a1y′ ¼ minð10,3Þ ¼ 3, a2y′ ¼ minð10−3,5Þ ¼ 5, and a3y′ ¼ minð10−3−5,6Þ ¼ 2. Under scenario y′, the optimal revenue of DUPI Ry′(10) ¼300  3 þ200  5 þ100  2¼ 2100 and the probability py′ ¼ P(N1t ¼3)  P(N2t ¼5)  P(N3t ¼6) ¼ 0.004359. With all the scenarios, the expected revenue of DUPI by selling 10 seats is E(Rt(10)|D¼ DUPI) ¼∑ypyRy(10) ¼2251.88 and, similarly, the expected revenue of DUPI by selling 9 seats is E(Rt(9)| D¼DUPI) ¼∑ypyRy(9)¼2102.38. Hence, the expected revenue gap E(Gt(10)|D ¼ERG)≈E(Gt(10)| D¼DUPI)¼ E(Rt(10)|D¼ DUPI)−E(Rt(9)|D¼ DUPI)¼2251.88–2102.38¼ 149.5. Since E(Gt(10)|D ¼ERG) is greater than f3(¼ 100), the airline should reject the current request, and the number of seats remains 10. 3.5.3.2. When time t 40.3245. Each time a customer arrives, the same decision procedure can be performed. The computation results for the rest of the arrival customers are shown in Table 1. The revenue generated by the decisions using ERG is $2400. Since the DUPI method can obtain the optimal revenue, we use it as a base measurement for the effectiveness of the other approaches. By knowing the numbers of customers of the three classes at the end of booking horizon and applying the greedy rule, we have the optimal decision under perfect information as also shown in Table 1. The optimal revenue is $2400. “The gap below DUPI” is defined as the difference in revenue between an approach and DUPI divided by the revenue from DUPI. Table 2 shows the gap below DUPI for FCFS, DSR, and ERG. In this example, ERG outperforms DSR, and DSR in turn outperforms FCFS. Furthermore, in this example, ERG generates the same decisions as DUPI, as can be observed in Table 1. The effectiveness of these approaches will be seen in the extensive experiments of Section 4.

4. Computational experiments and analysis Together with the approaches in the literature, there are seven approaches that can be applied to this decision problem. They are the first-come-first-served rule (FCFS), the right booking policy (RBP), and the left booking policy (LBP) by Haerian et al. (2006), the discrete-time dynamic programming model (DP) by Lee and

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Hersh (1993), the omnibus model (OM) by Lautenbacher and Stidham (1999), and DSR and ERG proposed in this study. This section presents the simulation experiments to compare these seven approaches. 4.1. Experiment design 4.1.1. Non-homogeneous Poisson arrivals In order to compare with the DP model, the assumptions adopted by the DP model (Lee and Hersh, 1993) in which the booking horizon is divided into a number of time intervals (or, briefly, interval) and each interval has a constant Poisson arrival rate is used. Thus, in this study, the booking horizon is divided into 10 equal-length intervals. However, these intervals are not the small decision periods where there is at most one customer in the DP model; in our DP experiment, the length of such a small decision period is computed by the formula suggested by Lee and Hersh (1993). To simulate demand fluctuations, this study assumes that the expected arrival rate of customers during the booking horizon increases linearly from the start of the booking horizon until the time point that has the highest expected arrival rate; then, the expected arrival rate decreases linearly to the end of the booking horizon. The highest expected arrival rate time point is specified by a fraction (denoted by η) of the length of the booking horizon. For example, in Fig. 3, η is 0.6; thus, the highest expected arrival rate is at the end of interval 6. The smaller/larger η, the earlier/later the peak of arrival rate; also, ηo0.5/η4 0.5 implies that there are more/less customer arrivals in the first half of the booking horizon than the second half. Since the expected arrival rate is a constant in an interval, which is one of the assumptions used in the experiment in the DP model, we use the average of the expected arrival rate of an interval in Fig. 3 (i.e., the expected arrival rate at the middle point of an interval) as the constant expected arrival rate of the interval, as shown in Fig. 4.

4.1.2. Random problem generation Each random problem includes a random number of customers who arrive at random times during the booking horizon. The following constant parameters are used to generate random problems.

445

1. 2. 3. 4.

The number of classes is 3(J). The number of initial available seats is 100(s0). The duration of the booking horizon is 100(h). The number of intervals is 10, and the length of each interval is 10. 5. The price for class 3 is 1000(f3).

In addition, the random problems are generated under the control of the following four factors: 1. Price slope (α) determines the prices for classes 1 and 2. The price of class 3 (f3) is a constant parameter (1000). The price for class j is calculated by fj ¼f3 þ α  (3−j), j¼ 1, 2. For example, if α¼500, f1 ¼1000 þ500  (3−1) ¼2000, and f2 ¼1000 þ500  (3 −2) ¼1500. The smaller/larger α, the smaller/larger price differences there are between classes. 2. Demand/supply ratio (β) is defined by the ratio of the expected total customer requests to the total available seats. If β is greater than 1, the expected total customer requests are more than the total available seats. 3. Relative ratios of customer classes (γ) is a set of values that represents the relative ratios of the customers in the three classes. For example, (2:3:4) represents that the ratios of the expected number of customers in classes 1, 2 and 3 are 2:3:4. 4. Relative location of highest arrival (δ) is a set of values that represents the highest expected arrival rate time point during the booking horizon of the three classes. For example, (0.8, 0.5, 0.2) represents the highest expected arrival rate time points (η) for class 1, class 2, and class 3 are at the 0.8, 0.5, and 0.2 time points of the total booking horizon, respectively. In (0.8, 0.5, 0.2), the highest arrival rate of class 3 customers is at 0.2 of the booking horizon (earlier portion) and that of class 1 customers is at 0.8 of the booking horizon (later portion). Therefore, (0.8, 0.5, 0.2) represents that more lower-fare class customers arrive early than late and more higher-fare class customers arrive late than early. Table 3 shows a summary of the four control factors and their values. Overall, there are a total of 625 (¼5  5  5  5) combinations of these four factors. Under each factor combination, 100 random problems are generated. Thus, there are a total of 62,500 random problems used to simulate the seven approaches. 4.2. Performance comparisons

Fig. 3. An example of the expected arrival rate determinations.

Expected arrival rate

Time Fig. 4. Constant expected arrival rate within an interval.

For each generated problem, the complete information on all the arrival customers is known; therefore, the optimal decision under perfect information (DUPI) can be obtained. Unlike most previous studies using FCFS as a base approach for measuring performance, this study uses DUPI as the base approach. The problem with using FCFS is that its performance varies much under different demand/supply ratios. The “gap below DUPI” can be calculated for each approach in each generated problem. Table 4 shows the Duncan test results of the gap below DUPI for the seven approaches under significant level 0.001. Among all the approaches, from 62,500 random problem experiments, ERG has the smallest average gap below optimal DUPI, and the average revenue difference is merely 0.81%. Also, from Table 4, we observe that the seven approaches are classified into seven distinct subsets. The gap below DUPI between each pair of two approaches is significantly different. “A oB” represents that the gap below DUPI of approach A is significantly lower than that of approach B; i.e., A outperforms B. According to the Duncan test, ERG oOM oDP oRBP oDSR oLBP oFCFS. That is, ERG outperforms all other approaches. OM and DP are the second and the third best approaches. DSR is not as good as these three approaches.

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7.0%

Factors

Values

Price slope (α) Demand/supply ratio (β) Relative ratio of customer classes (γ) Relative location of highest arrival (δ)

10, 250, 500, 750, 1000 1, 1.25, 1.5, 1.75, 2 (5:3:1), (4:3:2), (3:3:3), (2:3:4), (1:3:5)

average gap below DUPI

Table 3 A summary of the control factors and their values.

(0.9,0.5,0.1), (0.8,0.5,0.2), (0.7,0.5,0.3), (0.6,0.5,0.4), (0.5,0.5,0.5)

6.0% 5.0% 4.0% 3.0%

RBP DP OM DSR ERG

2.0% 1.0% 0.0%

1

1.25

1.5

1.75

2

demand/supply ratio Table 4 Duncan test for the seven approaches. Mean (%)

1

2

ERG OM DP RBP DSR LBP FCFS

0.81 1.82 2.03 2.27 2.49 10.20 11.05

****

3

4

5

6

7

4.5% **** **** **** **** **** ****

4.0%

RBP

3.5%

DP

3.0%

OM

2.5%

OM

DP

DSR

ERG

3.0% 2.5% 2.0% 1.5% 1.0% 0.5%

5:3:1

4:3:2

3:3:3

2:3:4

1:3:5

Fig. 7. Average gap below DUPI for various relative ratios of customer classes.

ERG

1.5%

4.0%

1.0% 0.5% 0.0%

RBP

3.5%

relative arrival rate

DSR

2.0%

4.0%

0.0%

10

250

500

750

1000

price slope Fig. 5. Average gap below DUPI for various price slopes.

The reason may be that DSR makes decisions by comparing the probabilities of fulfillment and rejection as correct decisions and without considering the price variation among the different fare classes. Assuming all the parameters but price are the same, in one case, the prices for the two classes are 1 and 2, respectively. In another case, the prices for the two classes are 1 and 10,000, respectively. Yet even if the price variations are very different in these two cases, the DSR will make the same decision. The insensitivity to price variation of DSR is the reason why it does not perform as well. 4.3. Performances under various factors Since FCFS and LBP perform significantly worse than the other approaches, we have deleted their results in the following graphical presentations for the four control factors. The charts of Figs. 5 through 8 show the average gap below DUPI of the approaches under four different control factors. From Fig. 5, the average gap below DUPI of all the approaches increases as the price slope increases. A large price slope makes the problem difficult. However, the largest difference in average gap for ERG among various price slopes is only about 1%. But, the largest differences for OM and DP are about 2.5% and 3%. Therefore, we can see that ERG is more robust than OM and DP under different price slopes. Fig. 6 shows that ERG has the largest average gap below DUPI when the demand/supply ratio is 1.25. But the largest difference in

average gap below DUPI

average gap below DUPI

4.5%

average gap below DUPI

Approach

Fig. 6. Average gap below DUPI for various demand/supply ratios.

3.5%

RBP

OM

DP

DSR

ERG

3.0% 2.5% 2.0% 1.5% 1.0% 0.5% 0.0% 0.9,0.5,0.1

0.8,0.5,0.2

0.7,0.5,0.3

0.6,0.5,0.4

0.5,0.5,0.5

relative location of highest arrival Fig. 8. Average gap below DUPI for various relative locations of highest arrival.

average gap below DUPI among the various values of the demand/ supply ratio is only about 0.5%. However, the largest differences for OM and DP are about 1.2% and 1.6%. Thus, similarly, ERG is more robust than OM and DP under different demand/supply ratios. It seems that, under ERG, the ratio of 1.25 makes the problem more difficult than under other ratios. When the demand and supply are about the same (the ratio equals 1), the problem may be relatively easy since almost all the demand can be satisfied. Further, when the demand is much more than the supply (the ratio is greater than 1.5), the problem may also be relatively easy since only higher class customers can be satisfied. From Fig. 7, we observe that ERG is very robust under different relative ratios of customer classes. The largest difference in average gap below DUPI among different values of relative ratios of customer classes is only about 0.2%. This factor seems not to affect the performance of ERG much. However, the average gaps for OM and DP increase as the number of higher class customers increases and the number of lower class customers decreases. The largest differences for them are both about 2%. Fig. 8 shows that the average gaps below DUPI for the best three approaches (ERG, OM, and DP) increase as the highest arrival

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rate time point of class 1 moves toward the end of the booking horizon and the highest arrival rate time point of class 3 moves toward the start of booking horizon. However, the difference in average gap is very small (around 0.6%) for ERG and relatively large (around 3%) for OM and DP. Again, ERG is the most robust among all approaches. Interestingly, when all three classes have the same highest arrival rate time point at the middle of the booking horizon, all three approaches have similar performances. The earlier the highest arrival rate time point of class 3 and the later the highest arrival rate time point of class 1, the more difficult (a larger gap below DUPI) it is for the best three approaches. Also, the more low class customers arrive earlier and the more high class customers arrive later, the better ERG performs than OM and DP. From the above four figures, we can see that ERG outperforms the other approaches and is very robust under the four control factors. The largest difference in average gap below DUPI under various problem conditions is only about 1%. However, OM and DP are not as robust as ERG under these factors, especially in the price slope and the relative location of the highest arrival rate. The largest differences in average gap below DUPI under various problem conditions for both OM and DP are about 3%. Hence, ERG is the most effective method in this study for the airline industry to make fulfillment-or-rejection decisions.

4.4. Verification of the ERG approximation ERG performs the best among all test approaches. This section presents an experiment to verify the conjecture, mentioned in Section 3.4.3, that the expected revenue gap under decision policy ERG, E(Gt(s)|D ¼ERG) is very close to that under DUPI, E(Gt(s)| D ¼DUPI). The four control factor values in Table 5 are used to generate random problems for the experiments in this subsection. Since the revenue by utilizing ERG policy to sell s seats from time t to the take-off time is a random variable (Rt(s)|D ¼ERG), we can use simulation to generate samples of the random variable. The parameters of current time t ¼0, available seats s ¼100, and the booking horizon h¼ 100 are used in this experiment. Given all the parameters, a random problem can be generated. Then, by simulating the ERG approach from time 0 to time 100 in the random problem, we have a random sample of R0(100)|D ¼ERG. Similarly, we can simulate again to obtain a random sample of R0(99)|D ¼ERG. Then, calculating the difference between the two random samples, we have a random sample for G0(100)|D ¼ERG. Given the same random problem, we can easily calculate the sample values for R0(100)|D¼ DUPI and R0(99)|D ¼DUPI by using the greedy rule. Then, a sample value for G0(100)|D ¼DUPI can be obtained. Now, we have a pair of sample values for G0(100)|D¼ ERG and G0(100)|D¼ DUPI. By repeating generating random problems, simulating ERG, and applying the greedy rule, we can have more samples for the two random variables—G0(100)|D ¼ERG and G0(100)|D ¼DUPI. In our experiment, a total of 100 random problems are generated and, thus, 100 samples of the two random variables are obtained. A t-test for the equality of the means of the two random variables, G0(100)|D¼ERG and G0(100)|D ¼DUPI, under significant level 0.001 is performed and the result shows that the means of Table 5 The values for the control factors in the experiment verifying ERG. Factors

Values

Price slope (α) Demand/supply ratio (β) Relative ratio of customer classes (γ) Relative location of highest arrival (δ)

500 1.5 (2:3:4) (0.8,0.5,0.2)

447

the two random variables are not significantly different. Such a result validates that replacing E(Gt(s)|D ¼ERG) by E(Gt(s)|D ¼DUPI) in Section 3.4.3 is a very good approximation. 4.5. Degree of optimality for allocation method in the ERG To compute an expected revenue, the optimal allocation method is utilized to optimize the decision under the condition that the perfect demand information of each scenario is deterministic and known. In the problem discussed in the present study, an optimal decision can be obtained using a simple greedy rule, in which the highest class is first fulfilled down to the lower classes until there no more seats remain. This subsection investigates the importance of using an optimal approach to solve the allocation problem under a given scenario. Two parts of the experiments are discussed in this subsection. One uses a sub-optimal method to obtain allocation decision and the other one uses a hyper-optimal method. The purpose of these experiments is to demonstrate the importance of using an optimal method to solve the allocation problem in a scenario with given perfect information and to investigate the performance when a non-optimal method is used. 4.5.1. An emulation of sub-optimal allocation method To emulate sub-optimum of allocation method, a sub-optimal allocation method is designed to calculate the allocation under a scenario. Under the condition that all seats will be fully utilized, the sub-optimal method will take away a number of seats from higher-class customers and assign the seats to lower-class customers, thereby obtaining lower revenue. In this experiment, the bth level sub-optimal method performs a downgrade allocation b times. For example, there are 5 available seats and the scenario is as follows: the number of customers of classes 1, 2, and 3 are 2, 3, and 3, respectively. The notation (x, y, z) is used to show that the number of seats allocated to classes 1, 2, and 3 are x, y, and z, respectively. Table 6 illustrates the allocations using various bth level sub-optimal methods under this scenario. In the optimal solution (b¼0), the allocation is (2, 3, 0). In the first-level suboptimal method (b ¼1), the allocation is (1, 3, 1). This is obtained by allocating a seat to a class 3 customer instead of a class 1 customer, which is a downgrade allocation from class 1 to class 3 because all class 2 customers are already fulfilled. In the secondlevel sub-optimal method (b¼ 2), the allocation is (0, 3, 2), which represents two downgrade allocations from two class 1 customers to two class 3 ones. In the third-level sub-optimal method (b¼ 3), three downgrades are performed—two from class 1 to class 3, and one from class 2 to class 3. Therefore, the allocation is (0, 2, 3). When b≥4, no downgrade can be further done without removing any available seat. In our designed sub-optimal method, a seat cannot be intentionally empty when there is a customer. Therefore, for bth level sub-optimal method where b≥3, the allocation is (0, 2, 3). The level (parameter b) of sub-optimal method represents the degree of optimality capacity for allocation method. The bth level sub-optimal method with s available seats is outlined as follows. Table 6 The allocations for various levels of sub-optimal methods. b

z1 ¼ 2, z2 ¼3, z3 ¼3

0 1 2 3 4 ⋮

(2, 3, 0) (1, 3, 1) (0, 3, 2) (0, 2, 3) (0, 2, 3) (0, 2, 3)

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4.5.1.1. Variables x the maximum number of downgrades that can be performed. qj the number of downgrades in class j. Step 1 If the number of all requests ð∑Jj ¼ 1 zj Þ is more than the available seats (s), proceed to the next step. Otherwise, go to step 6. h  i Step 2 Calculate x ¼ MIN ∑Jj ¼ 1 zj −s,b . Step 3 Let j ¼ 1. Step 4 Compute qj ¼ MIN(x,zj). Update zj ¼zj−qj and x ¼x−qj. Step 5 If x40, let j¼ jþ1 and go back to step 4. Otherwise, proceed to the next step. Step 6 Using the revised number of customers zj∀j, perform the greedy rule, as outlined in Section 3.2, to allocate the available seats. 4.5.2. An emulation of hyper-optimal allocation method Aside from the sub-optimal allocation method, a hyper-optimal allocation method, which emulates that certain constraints are relaxed, is also designed. Thus, the objective value of a hyperoptimal allocation method is higher than that of an actual optimal solution. To implement the hyper-optimal method, it is assumed that certain customers with allocated seats will be persuaded to pay the price of a higher class than the original class that they preferred. In this allocation method, the cth level hyper-optimal method performs a price upgrade c times; that is, it is assumed that customers with allocated seats are persuaded to pay the price of a higher class c times. For example, customers of class 2 can be price upgraded once to class 1, and customers of class 3 can be price upgraded once to class 2 or twice to class 1. Thus, given the numbers of customers in all classes, the total number of possible price upgrades is ∑Jj ¼ 1 ðzj  ðj−1ÞÞ. However, under the cth level hyper-optimal method, the maximum number of price upgrades will be c. Therefore, the number of price upgrades in the cth level hyper-optimal method is L ¼ MIN½½∑Jj ¼ 1 ðzj  ðj−1ÞÞ,c. In the experiments of this study, the relationship between the prices and class number is assumed to be linear; that is, the price differences between any two consecutive classes are equal and the constant price difference is α. Therefore, the revenue of the decision using the cth level hyper-optimal method can be calculated as ð∑Jj ¼ 1 f j zj Þ þ ðL  αÞ. In the experiments, the prices for classes 1, 2, and 3 are 300, 200, and 100, respectively; thus, the price differences (α) between any two consecutive classes is 100. It is assumed that there are four available seats and the number of customers of classes 1, 2, and 3 are 2, 1, and 2, respectively. Thus, in an optimal allocation, the number of customers with allocated seats is 2, 1, and 1 for classes 1, 2, and 3, respectively. Table 7 illustrates the revenues for various cth level hyper-optimal methods under this scenario. In the optimal solution (c ¼0), the revenue is (300  2) þ (200  1) þ (100  1) ¼900. In the first-level hyper-optimal method (c¼ 1), the revenue is 900 þ(1  100) ¼1000, which implies that one price upgrade is performed. Under this scenario, when c≥4, no upgrade can be further done. Therefore, for the cth level hyper-optimal method with c≥3, the revenue is 1,200.

4.5.3. Simulation results and analysis The parameters and control factors used to generate random problems are the same as the previous experiments. However, 20 random problems, instead of 100, are generated under each factor combination. Simulation experiments are conducted to compare 11 levels of sub-optimal and 11 levels of hyper-optimal methods. Table 8 shows the result of the Duncan test of the average gap below DUPI for the 11 levels of sub-optimal methods with an

Table 7 The revenue for various levels of hyper-optimal methods. C

z1 ¼2, z2 ¼ 1, z3 ¼ 1

0 1 2 3 4 ⋮

900 1000 1100 1200 1200 1200

Table 8 Duncan test for the 11 levels of sub-optimal methods. Method (b)

Gap mean (%)

1

1 0 2 3 4 5 6 7 8 9 10

0.9267 0.9503 1.0199 1.2530 1.5189 1.7724 2.0018 2.2088 2.4184 2.6705 2.9493

** **

2

3

4

5

6

7

8

9

10

** ** ** ** ** ** ** ** ** **

Table 9 Duncan test for the 11 levels of hyper-optimal methods. Method (c)

Gap mean (%)

1

0 1 2 3 4 5 6 7 8 9 10

0.9503 0.9788 1.0197 1.0628 1.1100 1.1587 1.2091 1.2583 1.3089 1.3606 1.4166

** **

2

** **

3

** **

4

** **

5

** **

6

** **

7

** **

8

** **

9

** **

10

** **

alpha value of 0.001. As can be seen in Table 8, the 11 levels of suboptimal methods are divided into 10 distinct subsets. The optimal method (b¼ 0) and the first-level approach (b¼ 1) are classified as the best subset. Based on these results, the optimality capability of the allocation method affects the performance of the ERG. The higher the b value is, the worse the performance of the ERG will be. Table 9 shows the result of the Duncan test of the average gap below DUPI for the 11 levels of hyper-optimal methods with an alpha value of 0.001. As can be seen in Table 9, the 11 levels of hyper-optimal methods are divided into 10 distinct subsets. The optimal method (c ¼0) and the first-level approach (c ¼1) are classified as the best subset. The higher the c value is, the worse the performance of the ERG will be. Fig. 9 shows the average gap below DUPI for the allocation methods with various degrees of optimality. As can be seen in the figure, the x-axis represents the degree of optimality for allocation method, where 0 represents the optimal allocation method (greedy method), integer −b represents the bth level sub-optimal method, and integer þc represents the cth level hyper-optimal allocation method. It is observed that the optimality capability of the method for a given scenario affects the ERG performance. Moreover, the better the allocation method is, the better will be the performance of the ERG.

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3.4.1, the number of branches is n(Uj) for class j; thus, the complexity of ERG is Οð∏Jj ¼ 1 nðU j ÞÞ, which may be a theoretically huge value. Several random problems with 1000 expected arrivals and 7 fare classes are generated to be calculated by ERG, and the average CPU time of an ERG computation is shorter than 0.25 s. Therefore, under the current generation of desktop CPU speed, ERG seems to be able to calculate in time for practical decision making.

5. Conclusion and future research

Fig. 9. Average gap below DUPI for a variety of allocation methods.

4.5.4. The reason for the effectiveness of the ERG approach Unlike traditional operation research approaches, the ERG does not attempt to model and solve the stochastic decision tree problem, as shown in Fig. 2. An opportunity cost is defined as the difference in the expected revenues, generated from future customers, between the two decisions—acceptance and rejection. Thus, comparing the opportunity cost with the valuation of the arrival customer, we will be able to make the decision. If the opportunity cost is higher than the price of the current customer, the customer should be rejected; otherwise, the customer should be accepted. As previously discussed, we are not able to calculate the opportunity cost for the decision tree in Fig. 2 when using the ERG approach. Instead, this study bypasses the difficulty of computing the opportunity cost of the decision tree for the ERG. If an optimal decision procedure were utilized, the opportunity cost could be calculated using the difference between two expected revenues. The first is an expected value of optimal revenue generated from an optimization method with perfect information under the condition that the customer is rejected— more seats are available for future customers. The second is the expected value of optimal revenue generated from an optimization method with perfect information under the condition that the customer is accepted—there is one less available seat for future customers. These two expected revenues are the optimal expected revenues that can be possibly achieved because the revenue is optimized under each scenario. Even though we do not know what the optimal decision method should be, in ERG approach we can compute the optimal expected revenues and make the decision based on such optimal expected revenues. If there is an optimal approach, the opportunity cost from it will be exactly equal to the revenue gap calculated using the proposed ERG approach. This means that the decision made from the decision tree problem in Fig. 2 using the ERG approach will be identical to that of an optimal approach, even when such approach is unknown. Thus, it could be said that “expecting the best gives us a good result”. 4.6. Complexity analysis The computer program is coded in Microsoft Visual Cþþ 6.0, and an Intel Core 2 Duo CPU 2.66 GHz personal computer with 2.00 GB RAM is used to run the experiments. The average computation times of all the decision procedures for an arrival customer are shorter than 0.1 s. A major concern about the application of ERG, performing the best in the above experiments, is the computation time since ERG needs to generate a tree of all the possible scenarios. From Section

Every time a customer arrives, the airline must immediately make a fulfillment-or-rejection decision. On one hand, if early requests are fulfilled regardless of the fare class, the airline may lose revenue from higher-fare customers arriving later. On the other hand, if the airline rejects too many lower-fare requests, it takes the risk of wasting capacity by taking off with empty seats. To solve such a stochastic trade-off problem, this study proposes and tests two heuristic decision approaches—the DSR decision policy and the ERG decision policy. By generating a scenario tree to represent all the possible scenarios when a customer arrives, the ERG policy calculates the expected revenues of the decision under perfect information, as well as the expected revenue gap. The ERG makes a fulfillment-or-rejection decision by comparing the expected revenue gap (opportunity cost) with the price of the class of the current customer that arrives. This study uses various factors to control the random problem generations for the simulation experiments. In these experiments, ERG outperforms all other approaches, including several approaches proposed in the literature. The average revenue generated by the ERG is only 0.81% lower than the optimal decisions with perfect information. In addition, the ERG is the most robust among all tested approaches under various problem conditions. An opportunity cost between the two options – acceptance and rejection – is needed to make a decision. To calculate an optimal opportunity cost, what we actually need are the two optimal expected revenues provided by an optimization approach, not an optimization approach itself. In the ERG policy, the optimal expected revenues are calculated without having an optimization decision approach for the decision tree. Using the results of expecting the best makes the ERG an outstanding policy. In order to make a comparison with existing approaches and to determine the effectiveness of various approaches, this study assumed that the acceptable price (fare class) of an arrival customer is known and that this serves as an input parameter to the decision policies. However, in an actual environment, an airline company should offer the prices of various fare classes that have available seats for sale. Since the ERG effectively compares the opportunity cost E(Gt(s)|D¼ ERG) with the price of a fare class, the set of fare classes that can accept a customer is determined by {j|fj≥E(Gt(s)|D ¼ERG)}. This indicates that only the fare classes with prices equal to or higher than the opportunity cost can be sold. In this study, the key input problem parameters are the Poisson arrival rates of various price classes that could be functions of future time. The focus of this study is to improve the decision method with known input parameters. However, in actual environments, the process of obtaining accurate input parameters of the problem is a major issue. Significant efforts are required to accurately forecast the future demands for effectively utilizing the proposed ERG approach that can significantly outperform the rest of the tested approaches. We assume that a customer who requests for more than one seat is approximated by a number of individual arrivals and that there are no cancellations and no noshows. Future studies could consider cases where a customer may

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simultaneously request multiple seats and where a customer may cancel a reservation. In addition, the dynamic pricing problem, which determines the fares for different classes, could be an interesting research problem. Given that the ERG approach is effective in solving the seat allocation problem of an airline company, this concept, after some modifications, may also be applicable to the rationing problems of other industries, including those related to inventory rationing or manufacturing capacity rationing. Acknowledgments This research was supported by a research grant (NSC 97-2221-E007-094-MY3) from the National Science Council of Taiwan, ROC and research projects (101N2073E1 and 102N2075E1) from National Tsing Hua University. References Belobaba, P.P., 1987. Airline yield management: an overview of seat inventory control. Transportation Science 21 (2), 63–73. Bertsimas, D., de Boer, S., 2005. Simulation-based booking limits for airline revenue management. Operations Research 53 (1), 90–106. Brumelle, S.L., McGill, J.I., 1993. Airline seat allocation with multiple nested fare classes. Operations Research 41 (1), 127–137. Curry, R.E., 1990. Optimal airline seat allocation with fare classes nested by origins and destinations. Transportation Science 24 (3), 193–204. de Boer, S.V., Freling, R., Piersma, N., 2002. Mathematical programming for network revenue management revisited. European Journal of Operational Research 137 (1), 2–92. Glover, F., Glover, R., Lorenzo, J., McMillan, C., 1982. The passenger-mix problem in the scheduled airlines. Interfaces 12 (3), 73–80.

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