Research Article

Joint optimization of airline pricing and fare class seat allocation Received (in revised form): 22nd June 2012

Claire Cizaire and Peter Belobaba Massachusetts Institute of Technology, Cambridge, MA, USA Claire Cizaire graduated from MIT with a PhD in 2011. This article presents part of her doctoral research. Dr Cizaire is the founder of Optimix, a start-up in the field of revenue management. She has also worked as a strategy consultant for Mars & Co in London. She holds an engineering degree from Supe´lec (Paris, France) and an MSc from MIT. Peter Belobaba is Principal Research Scientist at MIT, where he teaches graduate courses on The Airline Industry and Airline Management. He is Program Manager of MIT’s Global Airline Industry Program and Director of the MIT PODS Revenue Management Research Consortium. Dr Belobaba holds a Master of Science degree and a PhD in Flight Transportation Systems from MIT. He has worked as a consultant as a consultant on revenue management systems at over 40 airlines and other companies worldwide. Correspondence: Peter Belobaba, International Center for Air Transportation, Massachusetts Institute of Technology, 77 Massachusetts, Avenue, Room 33-318, Cambridge MA 02139, USA

ABSTRACT Although researchers have considered them as separate optimization problems, airline pricing and seat inventory control are interrelated and should ideally be considered as a single optimization problem. This article presents an approach for modeling the combined effects of pricing and fare class seat allocation on realized passenger demand, with the objective of solving for the optimal fares and fare class booking limits jointly and simultaneously. In our model, the underlying demand volume is a function of fares and we develop an approach for determining the impact of booking limits on the proportion of underlying demand that is accepted. On the basis of this insight, we formulate and solve a revenue maximization problem in which both fares and booking limits are the decision variables in a simplified problem with two fare classes and two booking periods. Journal of Revenue and Pricing Management (2013) 12, 83–93. doi:10.1057/rpm.2012.27 Keywords: pricing; revenue management; perishable products; joint pricing and seat allocation

INTRODUCTION Airline pricing and seat inventory control have generated a great deal of research attention but have typically been modeled and optimized separately. The prices of multiple fare products have traditionally been assumed to be a fixed and exogenous input to the revenue management (RM) seat allocation optimization problem. Yet, both processes

directly affect the consumer’s choice set and should ideally be considered as a single optimization problem, as first outlined by Weatherford (1997). By setting price levels and restrictions for each fare product, the airline pricing process defines the complete set of options that could be available to a passenger. However, RM booking limits on different fare products may

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render one or more of these options unavailable at the time of booking and therefore reduce the passenger’s actual choice set. In this article, the combined impact of both the prices and the booking limits on the realized demand is modeled and a joint optimization framework is proposed. The article continues in the following section with a short overview of the relevant literature. A two-product, two-period joint pricing and seat allocation problem is formulated in the subsequent section. We then propose an approach to solve this optimization problem. Finally, a numerical example is used to illustrate this approach and present a performance analysis. The last section summarizes the findings and presents possible directions for future work.

LITERATURE REVIEW A considerable body of work exists on airline pricing on the one hand and seat allocation optimization on the other. Whitin (1955) published a single period pricing model and Littlewood (1972) first considered the twoclass, single leg, capacity allocation problem. Belobaba (1987, 1989) examined the multiple fare product problem and proposed the expected marginal seat revenue heuristics (ESMRa and EMSRb). The EMSRb heuristic was widely adopted in airline RM systems as it provided reasonable approximations of optimal fare class booking limits and could realistically be implemented. Optimal solutions for singleleg flights were developed by Curry (1990), Wollmer (1992), Brumelle and McGill (1993) and Robinson (1995). With the growth of airline hub-and-spoke networks in the 1980s, RM researchers also developed origin-destination (OD) control mechanisms to account for network effects. Glover et al (1982) and Curry (1990) provided mathematical programming formulations for network revenue optimization. Smith and Penn (1988), Simpson (1989) and Williamson (1992) developed bid-price methods. McGill and van

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Ryzin (1999) provide a comprehensive overview of the development of airline RM methods. The literature on joint pricing and allocation models is, on the other hand, relatively scarce. It was only in the mid-nineties that the problem of joint pricing and seat allocation for perishable goods caught the attention of researchers. Weatherford (1997) first highlighted the importance of considering prices and suggested including them as decision variables in the seat allocation problem. He focused on the case of a single flight leg with multiple fare products, with expected demand for each fare product represented as a linear function of the product’s own fare as well as the next higher and lower fares. Only one time period was considered. Kuyumcu and Garcia-Diaz (2000) proposed an integer programming formulation to determine the best fare structure and booking limits given a restricted number of possible fare structures. However, the fares remained exogenous data. de Boer (2003) analyzed the joint optimization problem in a network. It was assumed that the demand for a product only depended on its own price. The analysis was limited to a single period, although heuristics were suggested for the multi-period problem. Cote et al (2003) proposed a model for jointly solving the pricing and seat allocation problem in a network with a competitor. The analysis was limited to a single period once again. In addition, the demand was assumed to be deterministic and the focus of the study was fare optimization. Chew et al (2009) developed a joint optimization approach for a single product with a two-period lifetime. The expected demand was a linear function of the fare. They suggested several heuristics for the multi-period problem. This article complements the existing body of work on joint pricing and seat allocation optimization. We focus on the single-flight leg case, but assume that the demands for the two fare products are dependent. Furthermore, we

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divide the selling period into sub-intervals to allow airlines to change their fares several times over the pre-departure booking period. Our model thus combines the joint pricing and seat allocation concepts of Weatherford (1997) and Chew et al (2009).

NOTATION AND ASSUMPTIONS We develop a model of a two-period pricing and allocation problem for two perishable products sharing a fixed inventory of seats. The scenario corresponds to a single carrier, single flight, single OD market environment. The flight has a fixed capacity of C seats. Two fare products, Fare Product 1 and Fare Product 2, are offered by the airline. The two products provide exactly the same in-flight services; nevertheless, these products are associated with two distinct sets of purchase restrictions and rules, and are therefore priced differently. In our notation, Fare Product 1 represents the more expensive, less restricted product. The two fare products remain available throughout the entire selling period unless the flight sells out. This approach to pricing has been implemented in recent years by several large airlines, and is typically referred to as ‘fare families’. These airlines offer and brand a limited number of differentiated fare ‘families’ with clearly defined restrictions and service amenities. Within each fare family, however, there can be numerous price levels that can vary by time to departure. The booking period is divided into two subintervals, TF1 and TF2. Bookings start to be accepted at the beginning of the first time frame, TF1. The prices of the two products can be modified at the start of each time frame. In the notation, xt and yt represent, for each TFt, the prices of Fare Product 1 and Fare Product 2, respectively. These price points are decision variables of our optimization problem. In addition, the airline can limit the total number of seats to be sold in the first time frame, thereby protecting seats for demand in the second time frame, which could be charged

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Table 1: Summary of notation

Indice

Description

C xt yt

capacity of the flight. price of Fare Product 1 in TFt price of Fare Product 2 in TFt. We impose that for all t, ytpxt booking limit corresponding to TF1 combined demand for Fare Product 1 and 2 in TFt expectation of the combined demand for Fare Product 1 and 2 in TFt bound of the uniform distribution in TFt probability that a random passenger chooses Fare Product 1 in TFt total revenues generated by the two fare products in TFt the total revenues generated by the sale of the two fare products over the entire booking period

z1 ntotal, t mtotal,t st pt Rt Rtotal

higher fares. The booking limit is noted z1. We assume that the total capacity is nested between the two time frames: unsold seats from the first time frame are available for booking in the second time frame. The flight capacity C limits the total number of bookings that can be accepted over the course of the two time frames. A summary of notation is provided in Table 1. The joint pricing and seat allocation problem modeled here consists of maximizing the total revenues generated by the sale of the two fare products during the two time periods by optimizing the four price points and the first time frame’s booking limit. We make the following assumptions regarding the demand formulation: 1. The demands in the two time frames are independent. 2. The total demand for both products in TFi, noted ntotal, i, is uncertain, and modeled as a stochastic additive function ntotal,i ¼ mi(yi) þ ei.

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3. The expectation of the total demand in TFi, denoted mi, is a linear function of the lower price:h mi(y i i) ¼ aibiyi, with ai, biX0 and yi 2 0; bai

i

4. The random variable ei is uniformly distributed: eiBU[sI, si]. 5. The probability that a passenger chooses the less restricted product is pi ðxi ; yi Þ ¼ 1=ð1 þ eai bi yi þci xi Þ, with bi, ciX0. The first two assumptions are widespread in the literature. Mills (1959, 1962), Nevins (1966), Lau and Lau (1988), and Chew et al (2009) have, for example, all assumed that the random factor is independent of the demand’s mean and that its impact is additive. The hypothesis that the demand is a linear function of price is also very common (Weatherford, 1997; Lau and Lau, 1988; Chew et al, 2009). It can be shown that the second part of the third assumption, on the lower available price, is a good approximation for large Fare Product 2 demand relative to Fare Product 1 demand (Cizaire, 2011). In the literature on RM, the most commonly used probability distribution of demand is the Gaussian distribution (Weatherford, 1997; Kuyumcu and Garcia Diaz, 2000). However, several reasons led us to choose the uniform distribution instead. As we shall see in the following section, determining the expected number of accepted bookings, or censored demand, is critical to the joint optimization problem. The probability density function of the sum of two independent random variables is the convolution product of their individual density functions. While the convolution of two unbounded Gaussian probability density functions is a simple Gaussian probability density function, there is no closed-form expression for the convolution of bounded Gaussian distributions. In other words, when the booking limit or the flight capacity truncates the probability distribution function of the demand in the first time frame, the convolution product becomes extremely complex when the

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distributions are Gaussian. The uniform distribution does not have this drawback. Furthermore, the approach we develop with the uniform distribution can be used to approximate the Gaussian distribution. The Gaussian function can be viewed as the limit of a sum of uniform functions. With the fifth assumption, we use a binary logit model to model the passengers’ choice alternatives. The consumer has two alternatives in his choice set: Fare Product 1 and Fare Product 2. The utilities of Fare Product 1 and Fare Product 2 in TFt are modeled as u1,t ¼ di,tctxt and u2,t ¼ d2,tbtyt, respectively. The resulting attractiveness of Fare Product i is eui;t and the probability that a passenger chooses Fare Product 1 over the other product is given by pi ðxi ; yi Þ ¼ eu1;t =ðeu1;t þ eu2;t Þ; The simplified probability function is pi ðxi ; yi Þ ¼ 1=ð1 þ eai bi yi þci xi Þ; where bt, ct represent the price sensitivities associated with Fare Product 1 and 2, respectively. The constant at is a measure of the relative intrinsic attractiveness of the lower fare product. The resulting demand for each fare product is a non-linear function of the two prices. For simplifying purposes, most of the previous studies addressing the multiple-product problem assumed that the demands for the different products are independent of each other (Kuyumcu and Garcia Diaz, 2000; de Boer, 2003; Cote et al, 2003; Chew et al, 2009). In reality, the demand for a product depends not only on the product’s own price, but also on the other product’s price: there is diversion between the products. A change in the price of one of the fare products affects not only its own demand but also the demands for the other products. It is therefore important in the joint pricing and seat allocation approach to take this dependence into account.

JOINT PRICING AND SEAT INVENTORY CONTROL MODEL The objective function in our model is the total revenue Rtotal generated by the sale of the

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two products over the course of the two time frames:

time frame, that is Cmin(ntotal,1, z1), as outlined in the following equation:

Rtotal ¼ R1 þ R2 with ( ntotal;1 ½p1 x1 þ ð1  p1 Þy1 ; if ntotal;1 oz1 R1 ¼ z1 ½p1 x1 þ ð1  p1 Þy1 ; otherwise   ( ntotal; 2 ½p2 x2 þ ð1  p2 Þy2 ; if ntotal; 2 oC  min ntotal;1 ; z1 R2 ¼    C  min ntotal;1 ; z1  ½p2 x2 þ ð1  p2 Þy2 ; otherwise

The two underlying demands ntotal,1 and ntotal,1 are uniformly distributed. The objective function for the stochastic model is therefore the total : expected value of the total revenue, R Z   total ¼ min ntotal;1 ; z1 ½p1 x1 þ ð1  p1 Þy1  R   f1 ntotal;1 dntotal;1 ZZ    þ min ntotal;2 ; C  min ntotal;1 ; z1 ½p2 x2 þ ð1  p2 Þy2      f1 ntotal; 1 f2 ntotal; 2 dntotal;1 dntotal; 2

For TF1, the expected revenue can 1 ðx1 ; y1 ; z1 Þ ¼ ½p1 x1 þ be rewritten as: R ð1  p1 Þy1  naccepted;1 where naccepted;1 is the expected value of the censored demand: naccepted;1 8 > < m1 ; ¼

z1 þm1 > 2

:

z1 ;

8 ntotal;2 ½p2 x2 þ ð1  p2 Þy2  > > > >   > < if ntotal;2 oC  min ntotal;1 ; z1 R2 ¼    > C  min ntotal;1 ; z1  ½p2 x2 þ ð1  p2 Þy2 ; > > > > : otherwise

The seat supply for the two time frames is nested: the physical constraint embodied by the flight capacity does not solely apply to bookings of a single time frame but to the combined bookings of both time frames. The booking limit in TF1 and the flight capacity affect the number of bookings that can be accepted in the two time frames, as shown in Figure 2. Probability Density Function of the TF1 Underlying Demand 1

if z1 Xm1 þ s1 

ðm1 z1 Þ2 4s1



s1 4

;

if z1 2 ½m1  s1 ; m1 þ s1  if z1 pm1  s1

The probability density function of the censored demand in TF1 depends on the booking limit imposed by the airline, as shown in Figure 1. The maximum number of bookings that can be accepted in the second time frame depends on the remaining capacity at the end of the first

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2 1

Underlying 1 − 1

1

1 + 1

Demand

Probability Density Function of the TF1 Censored Demand n accepted ,1

1 + 1 − z1 2 1

1 2 1

z1 1 − 1

1

Censored 1 + 1

Demand

Figure 1: Underlying and censored demands in TF1.

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underlying TF2 demand. For m1  s1 p C  m2  s2 pz1 ;

TF2 Underlying Demand ntotal,2 70

ntotal,2 = C– ntotal,1

naccepted; 2 "

60

# ðm2 þ s2  C þ z1 Þ3 ðz1  m1 þ s1 Þ ¼ m2  2s1 12s2 ðz1  m1 þ s1 Þ " # C  z1 þ m2 ðm2  C þ z1 Þ2 s2  þ  2 4s2 4

50 22

40

(1,2) 30



21

20

ðm1 þ s1  z1 Þ 2s1

For Cm2s2pm1s1pz1, the expected censored demand is

10 0 20

30

40

50

60

70

" naccepted; 2 ¼ m2 

Underlying Demand ntotal,1 Legend: Infeasible due to probability distribution Infeasible due to z1

ð m þ s 2 þ m1  s1  C Þ 2  2 4s2

Infeasible due to C



Figure 2: Constraints on the bookings in the two time frames.

The expected value of the revenues in TF2 is a function of the expected censored 2 ¼ demand in TF2, denoted naccepted; 2 :R ½p2 x2 þ ð1  p2 Þy2  naccepted; 2 The expected value of the total censored TF2 demand can be found geometrically. The region of possible values for ntotal, 1 and ntotal, 2 can be divided into four smaller regions by the constraints z1and Cs shown in Figure 2. For each of one these four regions, we can find the ordinate of the barycenter, and thus deduct naccepted; 2 We assume that the condition Cz1p m2 þ s2 is always satisfied. In order words, we assume that the fares are such that the total demand may exceed the flight capacity. If this is not the case, the total demand is much lower than the capacity and there is little need for seat allocation optimization. There are three cases to consider, depending on the position of the flight capacity constraint with respect to the

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#

ð z1  m 1 þ s 1 Þ 2s1 " # C  z1 þ m2 ðm2  C þ z1 Þ2 s2  þ  2 4s2 4



88

z21  ðm1  s1 Þ2 12s2

ð m 1 þ s 1  z1 Þ 2s1

For m1s1pCm2s2, we have naccepted; 2 ¼ m2 : The assumption that the demand follows a uniform distribution allowed us to determine the exact expression of the censored demand. The geometrical approach would not have been possible with a Gaussian distribution. Instead, one would have had to resort to simulations to estimate the number of accepted bookings. The objective function is a non-linear, neither concave nor convex, function. A nonlinear maximization technique such as Powell’s algorithm can be used to determine the optimal set of fares and booking limit.

PERFORMANCE ANALYSIS In this section, we illustrate our joint pricing and seat allocation approach with a numerical example. The assumed parameters for the demands in the two time frames considered are given in Table 2. The flight capacity is 100 seats.

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This numerical example is used to compare several different optimization approaches: K K

K

traditional leg-based RM (EMSRb); joint pricing and seat allocation optimization; joint pricing and seat allocation optimization with additional booking limits on Fare Product 2.

For each approach, we run Monte Carlo simulations in which the demand is stochastic. The total demand for the two fare products is generated at the beginning of each time period. We run two distinct types of simulations. In the first type of simulation, the demand is uniformly distributed in both time frames. In the second type, the demand is normally distributed in both time frames. We test the normal distribution in order to allow for a better comparison between the joint optimization and EMSRb leg-based seat allocation, which assumes that the demand is Gaussian. The demands for the two time frames are independent. For both demands, we generate 1000 samples. We use the parameters displayed in Table 2 to model the expected demands. The standard deviations of the simulated demands in TF1 and TF2 are 20 and 12, respectively. The corresponding bounds for the uniform distributions of demand are 35 and 21, as outlined in Table 2. For the normal distribution, the demand is truncated in order to prevent any instance of negative demand. Finally, the flight

Table 2: Parameters for the demand functions

TF1

TF2

Total demand

a1=135 b1=0.44

a2=85 b2=0.20

Probability

a1=0.864 b1=0.020 c1=0.009

a2=0.038 b2=0.016 c2=0.008

Bounds

s1=35

s2=21

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capacity does not affect the probability that a passenger chooses the higher fare product.

Traditional RM approach To illustrate the traditional RM approach, we use a fixed fare structure combined with a leg-based seat allocation method, the Expected Marginal Seat Revenue method, also noted EMSRb (Belobaba, 1992). The results of this standard RM approach will be used as a benchmark. In the traditional RM approach, the fares of the two fare products remain fixed throughout the entire booking period. We used the stochastic joint optimization approach to find the optimal pair of fares for EMSRb. In this formulation, the booking limit of the first time frame is equal to the flight capacity. The optimal pair of fares for TF1 and TF2 was calculated to be US$428 and $211. The predicted censored demand for Fare Product 1 and Fare Product 2, given the fares, is 24 and 62, respectively. The associated standard deviations are 11 and 13. The resulting EMSRb booking limit for Fare Product 2 is 76 (the booking limit of Fare Product 1 is the capacity, 100). In our simulation, the total demand for the two fare products is generated for each time frame, given the optimal pair of fares. We deduce the demand for each fare product based on the implied probabilities p1 and p2. The booking limit of 76 seats is then applied to demand for Fare Product 2 and the flight capacity limit is applied to the sum of the demands for the two products. The simulated average revenues are $24.9k and $25.0k for the uniformly distributed demand and the Gaussian demand, respectively. In both cases, the average load factor is 82.0 per cent. These results serve as a baseline for the remainder of the simulations in this section.

Stochastic optimal set of fares and booking limit The stochastic optimal solution to the joint pricing and seat allocation problem is shown in Table 3. On the basis of the assumed input parameters, we deduce the total number of accepted bookings and revenues implied by the

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chosen fares and booking limit, as shown in Table 4. The optimal fares and booking limit determined with the joint approach are expected to result in an average load factor of 82.6 per cent. The expected total revenue is $25.7k, higher than the results for the traditional RM approach. We run several sets of Monte Carlo simulations to compare the results of the joint model with the results obtained previously with EMSRb. Table 5 summarizes the simulation results. They are very close to the values predicted by the stochastic model in the uniform distribution case. The simulated average revenues are higher than predicted when the simulated demand is normally distributed. The stochastic model is based on the uniform distribution and the relative differences between the theoretical and simulated results with a Gaussian distribution are therefore slightly larger for the Gaussian distribution. The simulated average censored demands are very close to the predicted censored demands for both demand distribution types. In the case of the uniform distribution, the relative difference between the simulated censored demand and the predicted expected censored demand is 0.3 per cent and 1.2 per cent for TF1 and TF2, respectively. For the uniform demand

distribution, the simulated average total revenue is only 0.4 per cent higher than the predicted revenue of $25.7k. For the Gaussian distribution, the simulated average revenues are 1.0 per cent higher than predicted, due to the higher average censored demand. The Gaussian distribution gives more weight to the mean of the underlying demand. Thus, the expected censored demand for this distribution type lies between the uniform distributions’ naccepted and m. We can use the naccepted computed for the uniform distribution as a lower bound for the expected censored demand in the case of a Gaussian distribution. The comparison with the traditional RM approach is summarized in Table 6. The joint approach leads to similar but slightly higher load factors than the traditional RM method’s load factors. The revenues, on the other hand, are much larger. The joint approach provides more than 3 per cent increase in revenues from the traditional RM method, which already had ‘optimized’ fares. Even with the Gaussian distribution, which is the assumed distribution for the calculations in EMSRb, the stochastic joint optimization approach provides a 3.9 per cent increase in revenues.

Table 5: Average censored demands and revenues Table 3: Stochastic optimal solution to the joint problem

Fare Product 1 Fare Product 2 Booking limit

TF1

TF2

x1 =$383 y1 =$197 z1=73

x2 =$483 y1 =$237

TF1 Accepted bookings TF2 Accepted bookings Load factor (%) Total revenues (k$)

Uniform demand

Gaussian demand

48.5 34.2 82.8 25.8

48.1 35.0 83.1 25.9

Table 4: Implied censored demand and revenues

Average fare ($) Underlying demand Accepted bookings Revenues (k$)

90

TF1

TF2

Total

279.8 mtotal,1=49.5 ntotal;1 ¼ 48:7 1 ¼ 13:6 R

355.8 mtotal,1=37.6 ntotal;2 ¼ 33:9 2 ¼ 12:1 R

— mtotal=87.1 ntotal ¼ 82:6  total R ¼ 25:7

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Joint optimization of airline pricing and fare class seat allocation

Stochastic optimal fares and booking limit with an additional booking limit on Fare Product 2 The stochastic model we developed includes a single booking limit, which protects seats for the later time frame versus the earlier one, rather than for a higher fare product versus a lower fare product. In practice, airlines typically impose a booking limit on the lower fare product as well. We will use our numerical example to test this other type of booking limit on our model and show that this additional type of booking limit is redundant. Let z2, TF1 and z2, TF2 be the booking limits for the lower fare product in the first and second time frame, respectively. These limits are not part of the stochastic model’s output. We shall therefore test several levels for each one of them. The tested values for z2,TF1 are between

Table 6: Comparison between stochastic joint optimization and a traditional revenue management approach

Uniform demand

Gaussian demand

Accepted bookings: Traditional RM Stochastic Joint Opt. Difference

82.0 82.8 þ 1.0%

82.0 83.1 þ 1.4%

Revenues (k$): Traditional RM Stochastic Joint Opt. Difference

$24.9 $25.8 þ 3.4%

$25.0 $25.9 þ 3.9%

a slightly lower value than the underlying expected Fare Product 2 demand in TF1, given the optimal TF1 fare y1 , and the maximum demand for Fare Product 2 in TF1, which is defined by optimal TF1 booking limit z1 . For the second time frame, we test values between 17 that are slightly lower than the expected underlying demand for Fare Product 2 in TF2 and 45. The simulation results are summarized in Table 7. The additional booking limits led to a 3.5 per cent to þ 0.02 per cent change in revenues. For z2,TF1 ¼ 31, which is about 15 per cent higher than the expected number of booking requests for Fare Product 2 in the first time frame, we observe a slight increase in revenues ( þ 0.02 per cent) when z2,TF2 is large enough. All other combinations of booking limits led to a decrease in revenues. Our optimization model does not include booking limits on Fare Product 2. Therefore, enforcing such booking limits with the stochastic optimal set of fares and z1 is unlikely to be optimal or result in significantly higher revenues in our simulations. The objective of limits on lower fare products is to protect seats for passengers with a higher willingness-to-pay who usually arrive later in the booking process. The time dimension of this strategy is already taken into account in our model. By dividing the selling period into two time frames and imposing a booking limit on the first time frame, we ensure that seats are saved for the late, high-revenue passengers of TF2.

Table 7: Changes (%) in revenues due to the implementation of Fare Product2’s booking limits

z2TF2

z2TF1

41 37 34 31 27 24

17

21

25

29

33

37

41

45

2.7 2.4 1.6 1.5 2.3 3.5

1.8 1.5 1.3 1.1 1.3 1.8

1.3 1.0 0.8 0.8 0.9 1.3

0.1 0.0 0.2 0.6 0.5 0.2

0.5 0.2 0.1 þ 0.0 0.1 0.6

0.5 0.2 0.1 þ 0.0 0.1 0.6

0.5 0.2 0.1 þ 0.0 0.1 0.6

0.5 0.2 0.1 þ 0.0 0.1 0.6

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The fares, which are now decision variables, also enable us to control the demands for the two fare products in each time frame. These demands are assumed to be dependent on the two products’ fares and the stochastic model’s output therefore implicitly optimizes the mix of Fare Product 1 and 2 passengers. The potential revenue impact of additional booking limits on Fare Product 2 is thus negligible. Lastly, our model does not make any assumptions on the arrival order of Fare Product 1 and 2 passengers within each time frame. Contrary to Weatherford (1997) for example, we do not assume that all lower-revenue passengers arrive first and higher-revenue passengers last. If we were to assume such an arrival order within time frames, application of a booking limit on Fare Product 2 bookings would lead to larger increases in revenues.

CONCLUSIONS Pricing and RM decisions are complementary and interrelated, but have been considered sequentially both in the literature and in airline practice. In this article, we propose a new model to jointly optimize fares and booking limits for a two-product, two-period pricing and seat allocation problem. The demand is assumed to be a uniformly distributed random variable, with the mean being a linear function of the lower fare. We use a geometrical analogy to determine the censored demand of each time frame and express the new objective revenue function. The joint approach relies on both the prices of the two products and the booking limit to maximize the total revenues generated by the two fare products over the two time periods. A numerical example illustrates the advantages of this joint approach. Simulations confirm the benefits of accounting for the demand uncertainty and the constraints imposed on the demand in the problem formulation. The simulations showed that the proposed approach performs well when compared to a traditional RM approach, with fixed fares and a leg-based seat allocation method. The joint pricing and

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seat allocation solution provided a 3.4–3.9 per cent increase in revenues. Our model does not account for any network or competitive effects, but can still be used in practice as a joint pricing and RM guideline. This research could be further expanded to account for additional time frames, additional products or to take into account network effects. The model could also be used as a starting point to conduct more analysis on competitive effects, as one of the products could represent a competitor’s offer. The extension of this framework to the multiple fare product, multiple time period problem was described in Cizaire (2011) and will be the topic of a forthcoming paper. We have developed this joint pricing and seat allocation optimization model with reference to the airline industry, as our problem formulation was indeed inspired by the growing use by airlines of ‘fare family’ branding and pricing practices. As is the case with much of the research on pricing and RM developed for airlines, our model is clearly applicable to mostly any other industry in which the same concepts of product differentiation can be applied, and where the price levels associated with each product can be changed dynamically over multiple periods of the selling process.

ACKNOWLEDGEMENT This work was funded by the MIT PODS Consortium.

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