Advertisement
Browse Subject Areas
?

Click through the PLOS taxonomy to find articles in your field.

For more information about PLOS Subject Areas, click here.

  • Loading metrics

Open Access

Peer-reviewed

Research Article

Competitive analysis of online revenue management with two hierarchical resources and multiple fare classes

Abstract

Resource allocation problem is one of key issues in the field of revenue management. The traditional models usually rely on some restrictive assumptions about demand information or arrival process, which is sometimes out of line with reality. To overcome this shortcoming, the method of competitive analysis of online algorithms, which eliminates the need for the assumptions on demand and arrivals, is adopted to deal with the quantity-based revenue management problem. The current model in this paper considers two downgrade compatible levels of resources. Given the capacities and fares of both levels of resources, the objective is to accept appropriate customers and assign them to appropriate resources so as to maximize revenues. Compared with the existing literature, this paper generalizes the concerned resource allocation problem by considering multiple fares for each level of resources. From the perspective of online algorithms and competitive analysis, both an upper bound and an optimal online strategy are derived in this paper.

Citation: Ni G (2022) Competitive analysis of online revenue management with two hierarchical resources and multiple fare classes. PLoS ONE 17(10): e0276530. https://doi.org/10.1371/journal.pone.0276530

Editor: Ronnason Chinram, Prince of Songkla University, THAILAND

Received: July 10, 2022; Accepted: October 10, 2022; Published: October 24, 2022

Copyright: © 2022 Guanqun Ni. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the paper.

Funding: This work is partially supported by the MOE Layout Foundation of Humanities and Social Sciences under grant 20YJA630049 and Startup Program of Doctor Scientific Research in Shandong University of Technology under grant 722001. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The author has declared that no competing interests exist.

Introduction

In the field of revenue management (RM), one of the key issues is to deal with the quantity-based single-leg problem, where the decision maker allocates inventory of a kind of resource to a demand stream and each customer in the stream has a requested fare drawn from a given set of fare classes [1]. For example, given a flight with 100 available seats and three fare classes, e.g. $1000, $800, and $500, the airline has to decide how many seats to be allocated to different fare classes in order to maximize revenues. If the demand of each fare class is known in advance, the resource allocation problem is easy to solve, i.e., allocating seats to higher fare class as far as possible. Since the demand information is usually uncertain in practice, the majority of successful RM implementations rely heavily on accurate demand forecasting [2, 3]. Meanwhile, the traditional theoretical models often rely on some restrictive and unrealistic assumptions about demand information or arrival process such as independence and stationarity [4].

However, in the case of short life products or situations where demand history is not necessarily available, to obtain accurate characterization of demand is a challenge [5, 6]. Even in the fields of both airlines and hotels, where the application of RM techniques is the most successful, obtaining high quality forecasts from historical data is not easy. To relax the assumptions for both demand information and arrival process, we study the quantity-based RM problem from the perspective of online algorithms and competitive analysis [7]. This method eliminates the need for either of these assumptions, which is particularly suitable for decision makers who have to respond to events over time, like passengers booking tickets one at a time in the airline scenario.

Besides, the single-leg RM models in previous studies usually assume that the resource is non-hierarchical, which is sometimes out of line with reality. The quantity-based RM problem dealing with hierarchical resources is very common in practice. To see the importance of this scenario, consider an airline that provides potential passengers with two classes of tickets (i.e. hierarchical resources), economy class and business class, for one flight. The full fare of economy class ticket is undoubtedly lower than that of business class ticket. In order to increase revenue, airlines usually sell both economy class and business class at different discounts based on market segmentation. Generally, the lowest discount fare of business class is probably still higher than the full fare of economy class. Suppose that the set of all fares, including full fares and discount fares, is {f1, f2, …, fm} and f1 < f2 < … < fm. Without loss of generality, let fk (1 ≤ k < m) be the full fare of economy class and fm be the full fare of business class. Obviously, f1 and fk+1 are the lowest discount fares for economy class and business class, respectively. Note that, all fares are predetermined and invariable. Facing a request of customer with reservation price p for the ticket, if fi ≤ p < fi+1 for some i ≤ k, we call the customer economy class passenger and the airline can charge at most fi no matter which class, economy or business, is assigned to this customer. While, if fi ≤ p < fi+1 for some i ≥ k + 1 (specially, fm+1 = +∞ when i = m), we call the customer business class passenger and the airline has two choices, assigning the customer to business class charging fi or assigning the customer to economy class charging fk. Given the capacities of both economy class and business class, the airline has to make decisions on assigning how many customers with different reservation price to each class. Since the airline cannot know demand information in advance, the decision-making problem is indeed of an online fashion. Other notable applications are hotel booking with heterogenous rooms and car rental with different classes of autos. The major issue in these scenarios is how to accept appropriate customers and assign them to appropriate resources so as to maximize revenues, i.e., the payments of all accepted customers.

Our contributions can be summarized as follows. We propose a new online strategy for the quantity-based RM problem with two levels of resources where the resources are downgrade compatible. The current model in this paper generalizes the concerned problem by considering multiple fares for each level of resources. From a practical point of view, it is difficult, if not impossible, to exactly know future demand; that is the decision maker has to make resource allocation decisions in a learning-by-doing manner based on the previous decisions and outcomes. For this learning-by-doing decision-making problem, we focus on online strategy and use standard competitive analysis to evaluate an online strategy. In this paper, an upper bound of competitive ratio for the problem is derived. We also propose an optimal online strategy whose competitive ratio matches the upper bound.

The quantity-based resource allocation problem has long been studied in the RM literature. Littlewood was the first to formulate the single-leg RM problem with two fare classes [8]. He assumed that resource is sold in a low-before-high (LBH) manner; that is, demand in lower fare class arrives earlier. Given the probability distribution of demand for the high fare class, Littlewood determined the booking limit for the low fare class. With the same assumptions, Belobaba extended Littlewood’s model to multiple fare classes [9, 10]. The LBH assumption is widely applied in the traditional single-leg RM studies [11–13]. There are several other studies on the single-leg RM problem with various assumptions on demand or arrivals, e.g. arriving at non-overlapping intervals [14], a stochastic process [15], discrete time Markov Decision Process [16], or demand information on the choice behavior [17]. In practice, it is probably difficult to meet these assumptions on demand. Therefore, we need more robust approaches which do not rely heavily on demand information to deal with the quantity-based RM problem.

In the new stream of research, the works of [18–20] are the most related to our study, all of which adopt the approach of competitive analysis of online algorithms. Ball and Queyranne, to the best of our knowledge, were the first to deal with the single-leg resource allocation RM problem from the perspective of online algorithms and competitive analysis [18]. They defined the lower bounds on the best-possible algorithmic performance and designed optimal nested protection-level policies whose competitive ratios match the lower bounds. In their model, there is no any assumption on demand or arrivals. Also from the perspective of competitive analysis of online algorithms, Lan et al. studied the classical multiple fare single-leg RM problem with limited demand information where the decision maker only knows the lower and upper bounds of demand for each fare class [19]. That is, the work of [19] is more general than that of [18] by assuming the availability of limited demand information. The competitive analysis of online algorithms has also been applied for other single-leg RM problem under various situations in the literature [21–23]. All of these studies assume that the resource is non-hierarchical, which is sometimes out of line with reality. Like the example in the field of airlines, the quantity-based RM problem dealing with compatible hierarchical resources is very common. To overcome this shortcoming, in this paper we consider the single-leg RM problem with two levels of resources, which are downgrade compatible; that is the resource of high level can be used as low level, but the resource of low level cannot be used as high level. Recently, Ni et al. proposed an optimal online strategy for the simplest problem with two levels of resources, in which there is only one fare class for each level of resources [20]. However, taking the field of airlines as an example, there usually are multiple fare classes even for the same level of resources. In order to be more realistic, we further extend the work of [20] by considering more than one fare class for the same level of resources. It is worth noting that our model will degenerate to the Multiple-Fare Classes problem studied in [18] when there is only one level of resources available. Besides, the current model will degenerate to the 2-level-2-class problem studied in [20] if there is only one fare class for each level of resources.

The rest of this paper is organized as follows. In the following section, we introduce the problem and describe our approach. Next, an upper bound and an optimal online strategy are presented from the perspective of online algorithms and competitive analysis. Finally, applications and future directions are discussed.

Problem definitions

Consider the problem of a firm having two levels of resources, denoted by R1 and R2, to provide a kind of service for heterogeneous customers. Suppose that resource Ri has a capacity of Li (i = 1, 2) where Li is an integer, and each customer only needs one unit capacity of resource. Although both levels of resources can satisfy customers’ basic need, we assume that R2 serves customers better or more comfortably than R1; that is, the level of R2 is higher than R1. As described in the airline scenario, the firm indeed sells both levels of resources at different discounts based on market segmentation and predetermines the fare of each class. Let {f1, f2, …, fm} be the set of all fares and f1 < f2 < ⋯ < fm. Without loss of generality, let fk (1 ≤ k < m) be the full fare of unit R1 and fm be the full fare of unit R2. Facing a request of customer with reservation price p for unit resource, if fi ≤ p < fi+1 for some i ≤ k, the firm can charge at most fi no matter which level of resources, R1 or R2, is assigned to this customer. While, if fi ≤ p < fi+1 for some i ≥ k + 1 (defining fm+1 = +∞ for completeness), the firm has two choices, assigning the customer to R2 charging fi or assigning the customer to R1 charging fk. According to their reservation prices, customers are also divided into m classes, denoted by class Ci (i = 1, 2, …, m), and the reservation price of customers of class Ci is within the range of [fi, fi+1). Given the capacity of each level of resources, the objective of the firm is to choose the most profitable customers and assign them to the right resources to maximize the resulting revenue. All of the main symbols that are used in the resource allocation model are listed in Table 1.

thumbnail
Download:
Table 1. The symbols of the resource allocation model.

https://doi.org/10.1371/journal.pone.0276530.t001

If the demand information is known in advance, then it is an offline problem and there is an intuitional revenue-maximization strategy, i.e., assigning customers with higher reservation price to higher level of resources as far as possible. Namely, an optimal offline strategy first assigns customers of class Cm to resource R2 and then to resource R1 until either all customers of this kind are assigned or all resources are used. If there are some resources available after accepting all customers of class Cm, then accept and assign customers of class Cm−1 first to R2 as far as possible and then to R1, and so forth. This continues until all customers are assigned or all resources are used.

As mentioned in the introduction, we focus on an online version of the RM problem, wherein customers ask for the service one at a time and the decision on whether to accept the request as well as assigning it to which level of resources has to be made irrecoverably on its arrival without the information of further customers. For a revenue-maximization problem, the theoretical performance of an online strategy is measured by the ratio between the objective revenue achieved by the online strategy and the revenue achieved by an optimal offline strategy, which is called competitive ratio in the literature. Let ΩA be the set of all possible input sequences to an online strategy A. For ∀I ∈ ΩA, let VA(I) be the objective revenue achieved by A facing input I and let VOPT(I) be the objective revenue achieved by an optimal offline strategy. The competitive ratio of A is defined as . An online strategy A* is the best one if the competitive ratio of A* is cA* = supcA, where supcA is called the upper bound of the competitive ratio.

Competitive analysis for the online RM problem

This section gives detailed competitive analysis for the online RM problem with two levels of resources and multiple fare classes. First, an online strategy is designed in the following, and then an upper bound for the proposed online RM problem is derived.

An online nested protection strategy

For notational convenience, let t = L1/L2 be the ratio of the capacities and ri = fi/fi+1 be the ratio of the fares for 1 ≤ i < m in the following. Virtually set r0 = 0 and rm = 1. We define . Because 0 ≤ ri < 1 and t > 0, the denominator is larger than 1 and obviously 0 < α < 1.

For customers of class Cj, set if 1 ≤ j ≤ k; otherwise if k + 1 ≤ j ≤ m, set . Define Θ0 = 0 virtually. The following lemma can be derived from the definition of Θj.

Lemma 1 Given the definition of α and the values of Θj as defined,

(1) for 1 ≤ j ≤ k, and

(2) for k + 1 ≤ j ≤ m.

Proof of Lemma 1. For part (1) with 1 ≤ i ≤ j ≤ k, Θi − Θi−1 = α(L1 + L2)(1 − ri−1) directly from the definition of Θj, and thus

That means part (1) is straightforward and independent of the specific value of α. The reasoning of part (2), however, is non-trival. Firstly, considering the part of with k + 2 ≤ j ≤ m,

From part (1), we have that . By the definitions of Θk+1 and Θk, because of rm = 1,

By the definition of Θj for k + 2 ≤ i ≤ j ≤ m,

And thus,

Now, we can derive the left side of the equation in part (2) as follows.

Note that when j = k + 1, and we can similarly derive the result of part (2) for j = k + 1. Thus, the proof of part (2) is completed, which gives us the lemma.

Upon an arrival of any customer, denote by li, li1, and li2 the number of customers of class Ci that have been accepted by the online strategy and been assigned to resource R1 and resource R2, respectively. Let li = li1 = li2 = 0 for 1 ≤ i ≤ m initially. Now, we present an online nested protection strategy, denoted by ONPS, for the problem as follows.

Online nested protection strategy ONPS.

Upon an arrival of a customer of class Cj with 1 ≤ j ≤ k, accept the customer’s service request and assign the accepted customer to resource R1 if for any j ≤ h ≤ k and , or assign the accepted customer to resource R2 if for any j ≤ h ≤ k, , and for any k + 1 ≤ h ≤ m; otherwise reject it.

Upon an arrival of a customer of class Cj with k < j ≤ m, accept the customer’s service request and assign the accepted customer to resource R2 if for any j ≤ h ≤ m, or assign the accepted customer to resource R1 if for some h ∈ [j, m] and ; otherwise reject it.

By the above description of strategy ONPS, it guarantees a reservation capacity for each class of customers, and all the reservation capacities are nested. More precisely, for the highest j (≤k) classes of customers whose reservation prices are no more than fk, i.e. Ck−j+1, Ck−j+2, …, Ck, strategy ONPS defines a protection reservation capacity equal to Θk − Θk−j. With a little difference, for the highest j (<m − k) classes of customers whose reservation prices are more than fk, i.e. Cm−j+1, Cm−j+2, …, Cm, strategy ONPS defines a protection reservation capacity equal to L2 − Θm−j. In special, for the highest m − k classes of customers, i.e. Ck+1, Ck+2, …, Cm, strategy ONPS defines a protection reservation capacity just equal to L2 − (Θk − L1) = L1 + L2 − Θk. Notice that, applying ONPS, only the higher class of customers can occupy the protection capacity reserved for the lower class. The lower class of customers cannot occupy the protection capacity reserved for the higher class.

Theorem 1 For the online quantity-based RM problem with two levels of resources and multiple fares, strategy ONPS has a competitive ratio of α.

Proof of Theorem 1. Given any customer input instance I, let be the total number of customers of class Cj for all j ≤ i that are accepted by ONPS, and virtually set . Define where θi = Θi for 0 ≤ i ≤ k and θi = Θi + L1 for k + 1 ≤ i ≤ m. Because ONPS considers eligible resource from R1 to R2 (or, from R2 to R1) when assigning the accepted customers of class Cj for all j ≤ k (or, for all j > k), the definition of u implies that all the customers of class Cj in instance I with u < j ≤ m are accepted by ONPS and assigned to the right resources resulting in as much revenue as possible. Let Von and Vopt be the total revenue obtained by ONPS and by an optimal offline strategy OPT facing instance I, respectively. According to the status of u, there are four cases as discussed below.

Case 1. u = 0. This condition implies that all the customers in I are accepted by ONPS and assigned to the right resources from the above analysis, and thus Von = Vopt.

Case 2. 1 ≤ u ≤ k. For this case, let Π be the total revenue of the accepted customers each of which is of a reservation price strictly larger than fu, and ϕ be the total number of these customers accepted by ONPS. We already know that strategy ONPS reserves a capacity of Θk − Θu for all customers of classes Cu+1, …, Ck where 0 ≤ u ≤ k − 1. Hence, where the second inequality is due to Lemma 1. OPT at best accepts all the ϕ customers with revenue larger than fu as ONPS does, and accepts the rest L1 + L2 − ϕ customers with unit revenue at most fu, implying Vopt ≤ Π + fu(L1 + L2 − ϕ), and thus .

Case 3. k < u < m. In this case, Π and ϕ are defined as the same with Case 2. Similarly, since ONPS reserves a capacity of L1 + L2 − Θk for all customers of classes Ck+1, …, Cm, and a capacity of L2 − Θu for all customers of classes Cu+1, …, Cm, where k + 1 ≤ u ≤ m − 1, it is obvious that where V = 0 for u = k + 1 and for u > k + 1. By Lemma 1, Von ≥ α(fkL1 + fuL2) + Π. For OPT, it at best accepts all the ϕ customers with reservation price larger than fu and accepts the rest L2 − ϕ customers with reservation price at most fu for resource R2. Each customer assigned to resource R1 is of unit revenue at most fk. Therefore, Vopt ≤ Π + fu(L2 − ϕ) + fkL1, and thus .

Case 4. u = m. Similar to the analysis in Case 3, by Lemma 1, . For OPT, it at best gains a maximum revenue of Vopt = fkL1 + fmL2, and thus .

Combining the four cases, the revenue achieved by strategy ONPS is always no less than α times of that achieved by an optimal strategy, which gives us the theorem.

An upper bound

In this section, we present an upper bound of competitive ratio for the online RM problem with two levels of resources and multiple fares as stated in Theorem 2, which implies that the online strategy ONPS proposed in the previous section is an optimal one.

Theorem 2 For the online quantity-based RM problem with two levels of resources and multiple fares, any online strategy has a competitive ratio of at most α.

Proof of Theorem 2. In order to derive a bound on the best-possible performance for an online strategy for the problem, we use the following extreme instances Ii (i = 1, …, m): in instance Ii a sequence of i(L1 + L2) customers, namely, L1 + L2 customers of each class C1, C2, …, Ci arrive in this order. The optimum revenues earned by an offline strategy OPT are VOPT(Ii) = fi(L1 + L2) for 1 ≤ i ≤ k and VOPT(Ii) = fkL1 + fiL2 for k + 1 ≤ i ≤ m, respectively.

For any online strategy ON, let be the number of customers of class Ci accepted by ON when presented with instance Im. Note that ON has no way of knowing whether it is facing instance Ii or some Ij with j > i before it has seen the first i(L1 + L2) customers in the stated sequence. Therefore, for all 1 ≤ j ≤ i, ON will accept exactly customers of class Cj when presented with instance Ii. Let ωi = Θi − Θi−1 for i ≠ k + 1 and ωk+1 = Θk+1 + L1 − Θk where Θi is defined as the proposed online strategy ONPS. If for all i = 1, 2, …, m, then strategy ON is equivalent to ONPS, which implies that the competitive ratio of ON is just α by Theorem 1. Otherwise, if there exists some , letting , then there are two cases as discussed below according to the status of v.

Case 1. . In this case, to generate the worst algorithmic performance, the extreme instance Iv will be presented for ON. Obviously, the revenue gained by ON facing Iv is . Therefore, the best-possible performance of strategy ON is . Note that VOPT(Iv) = fv(L1 + L2) for 1 ≤ v ≤ k and VOPT(Iv) = fkL1 + fvL2 for k + 1 ≤ v ≤ m. By Lemma 1 and the definition of ωi, for any v, and thus .

Case 2. . Since ωm ≤ L2 = Θm, it is obvious that v < m in this case. Thus, there must be some h (0 ≤ h < m) satisfying for i = v, v + 1, …, v + h while . The rest reasoning is similar to Case 1. Presenting the extreme instance Iv+h+1 for ON, it is also that in this case.

Combining Case 1 and Case 2, it is always that VON ≤ αVOPT, which gives us Theorem 2.

Application discussion

In this section, we present a numerical example in the field of airlines to illustrate the results of our model and to discuss the application of the proposed theoretical model. In fact, our method and model can also be applied in other fields, such as hotel booking with heterogenous rooms and car rental with different classes of autos.

Consider an airline that provides potential passengers with 80 economy class seats and 20 business class seats, for one flight. The full fares of economy tickets and business tickets are predetermined equal to $1000 and $2000, respectively. Additionally, there are two discount fares of economy tickets respectively equal to $500 and $750, and there is only one discount fare equal to $1600 for business seats. Following the theoretical model setting, it is that L1 = 80, L2 = 20, and thus t = L1/L2 = 4. Meanwhile, f1 = $500, f2 = $750, f3 = $1000, f4 = $1600, and f5 = $2000. According to the definition of the proposed online strategy, we further have r0 = 0, r1 = 2/3, r2 = 3/4, r3 = 5/8, r4 = 4/5, r5 = 1, and thus α = 0.59, which means that any online seat allocation strategy cannot achieve a competitive ratio larger than 0.59 for this example.

In order to illustrate the results of our proposed model and online strategy, consider an extreme LBH customer arrivals stated as follows. In this LBH arriving process, a sequence of 100 customers of each class C1, C2, C3, C4, C5 arrive in this order. Obviously, the optimal offline strategy with full information would only accept 100 customers of class C5 and assign 20 of them to business seats charging f5. The rest 80 customers are assigned to economy seats at f3. The resulting maximum revenue is equal to Vopt = 20f5 + 80f3 = $120000. For the same customer arriving sequence, the first-come-first-served strategy would accept 100 customers of class C1 gaining $50000.

Following the online strategy ONPS proposed in our model, obviously Θ1 = 59, Θ2 = 79, Θ3 = 93, Θ4 = 18, and Θ5 = 20. That is, for the highest business class C5, the proposed strategy ONPS defines a protection reservation capacity of 2 business seats. For the highest two business classes of customers, i.e. C4 and C5, 7 business seats are reserved as the protection capacity. For the highest economy class C3, strategy ONPS defines a protection reservation capacity equal to 93-79 = 14. For the highest two economy class C2 and C3, strategy ONPS defines a protection reservation capacity equal to 93-59 = 34. For the lowest class C1, ONPS sets a maximum number of accepted customers equal to 59. Therefore, facing the same instance, the proposed strategy ONPS will accept 59 customers of class C1 and 20 customers of class C2, and assign them to economy seats. 14 customers of class C3 will be accepted and 13 of them will be assigned to business seats. The strategy ONPS will also accept 5 customers of class C4 and 2 customers of class C5, and assign them to business seats. The resulting revenue is equal to 59f1 + 20f2 + 14f3 + 4f4 + 2f5 = $70500, which is more than that gained by the first-come-first-served strategy.

Obviously, the revenue gained by ONPS is 0.59 (= 70500/120000) times of that gained by an optimal offline strategy. Note that, the competitive ratio of the proposed strategy is just equal to α = 0.59, which means that the presented extreme LBH instance is one of the worst cases for strategy ONPS from the perspective of competitive analysis of online algorithms. In other words, the practical performance will be better when facing other customer arriving instances.

Notice that, there is an implicit assumption on customer’s reservation price in our model. We assume that the decision maker knows the customer’s reservation price when she/he arrives. For some firm, in fact, it might analyze the psychological expectation of customer and draw the reservation price using big data technology. Even in the case where the firm cannot know customer’s reservation prices, our model can also be used to guide the decisions on how many resources to be sold at each fare, especially to set limits for lower fare classes and to reserve capacity for higher fare classes, like the applications of theoretical models implied in [18].

Conclusion

This paper studies an online revenue maximization problem of assigning heterogeneous customers to two levels of resources with capacity constraints. From the perspective of competitive analysis, an upper bound for this online revenue management problem is derived and an online strategy whose competitive ratio matches the upper bound is also proposed. The model proposed in this paper generalizes the 2-level-2-class problem studied in [20] by considering multiple fare classes for each level of resources, i.e., the strategy as well as the theoretical result will degenerate to that of [20] if k = 1 and m = 2 in the current model. Besides, it will degenerate to the single-leg-multiple-fare problem studied in [18] when there is only one level of resources available.

There are several directions for future research. First, it is interesting to study the general problem with more than two levels of resources. Second, it is valuable to investigate the scenario with limited downgrading for customers with high reservation price. Third, it is also interesting and challenging to consider resources allocation problem together with pricing decisions.

Acknowledgments

The author would like to thank the editor and four anonymous referees for their helpful comments and suggestions.

References

  1. 1. Talluri KT, van Ryzin GJ. The theory and practice of revenue management. New York: Springer-Verlag.; 2005.
    • 2. Boyd EA, Bilegan IC. Revenue management and e-commerce. Manag Sci. 2003 Oct;49(10):1363–1386.
    • 3. McGill JI, van Ryzin GJ. Revenue management: research overview and prospects. Transp Sci. 1999 May;33(2):233–256.
    • 4. Besbes O, Zeevi A. Dynamic pricing without knowing the demand function: risk bounds and near-optimal algorithms. Oper Res. 2009 Dec;57(6):1407–1420.
    • 5. Fisher M, Raman A. Reducing the cost of demand uncertainty through accurate response to early sales. Oper Res. 1996 Feb;44(1):87–99.
    • 6. Kurawarwala AA, Matsuo H. Forecasting and inventory management of short life-cycle products. Oper Res. 1996 Feb;44(1):131–150.
    • 7. Borodin A, El-Yaniv R. Online computation and competitive analysis. Cambridge: Cambridge University Press; 1998.
      • 8. Littlewood K. Special Issue Papers: Forecasting and control of passenger bookings. J Revenue Pricing Manag. 2005 Apr;4(2):111–123.
      • 9. Belobaba PP. Airline yield management: An overview of seat inventory control. Trans Sci. 1987 May;21(2):63–73.
      • 10. Belobaba PP. Application of a probabilistic decision model to airline seat inventory control. Oper Res. 1989 Apr;37(2):183–197.
      • 11. Curry RE. Optimal airline seat allocation with fare classes nested by origins and destinations. Trans Sci. 1990 Aug;24(3):193–204.
      • 12. Wollmer RD. An airline seat management model for a single leg route when lower fare classes book first. Oper Res. 1992 Feb;40(1):26–37.
      • 13. Brumelle SL, McGill JI. Airline seat allocation with multiple nested fare classes. Oper Res. 1993 Feb;41(1):127–137.
      • 14. Robinson LW. Optimal and approximate control policies for airline booking with sequential nonmonotonic fare classes. Oper Res. 1995 Apr;43(2):252–263.
      • 15. Lee TC, Hersh M. A model for dynamic airline seat inventory control with multiple seat bookings. Trans Sci. 1993 Aug;27(3):252–265.
      • 16. Lautenbacher CJ, Stidham S Jr. The underlying Markov decision process in the single-leg airline yield-management problem. Trans. Sci. 1999 May;33(2):136–146.
        • 17. Talluri KT, van Ryzin GJ. Revenue management under a general discrete choice model of customer behavior. Manag Sci. 2004 Jan;50(1):15–33.
        • 18. Ball MO, Queyranne M. Toward robust revenue management: Competitive analysis of online booking. Oper Res. 2009 Aug;57(4):950–963.
        • 19. Lan Y, Gao H, Ball MO, Karaesmen IZ. Revenue management with limited demand information. Manag Sci. 2008 Sep;54(9):1594–1609.
        • 20. Ni G, Zheng F, Xu Y. Competitive analysis of online revenue management with hierarchical resources. Inf Process Lett. 2019 Feb;142:41–45.
        • 21. Lan Y, Ball MO, Karaesmen IZ. Regret in overbooking and fare-class allocation for single leg. Manuf Serv Oper Manag. 2011 Jan;13(2):194–208.
        • 22. Lan Y, Ball MO, Karaesmen IZ, Zhang JX, Liu GX. Analysis of seat allocation and overbooking decisions with hybrid information. Eur J Oper Res. 2015 Jan;240(2):493–504.
        • 23. Ni G, Luo L, Xu Y, Xu J. Optimal online markdown and markup pricing policies with demand uncertainty. Inf Process Lett. 2015 Nov;115(11):804–811.