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An Optimal Control Problem of Dynamic Pricing
Abstract
An optimal control problem of dynamic pricing is studied. In the model, prices are chosen to sell multiple products to multiple customer classes over time. The products share a number of scarce resources. All parameters, such as the arrival rates of customers, their purchasing probabilities, their cancellation rates, and the cancellation refunds, are allowed to be time dependent. A solution method for the problem is developed, and is tested with some numerical examples. # Supported, in part, by the National Science Foundation under grant DMI-9875400. 1
... приложения для нескольких типов одновременно продаваемой продукции (Галего и Ван Ризин (1997) [9], Клейвет (2001) [10]). ...
У цій статті проведено аналіз сучасних методологічних підходів до дослідження проблем динамічного ціноутворення, виявлено їх обмеження і критичні питання щодо практичної реалізації. Більшість робіт передбачає безперервний перегляд цінової політики та обізнаність продавця у функції попиту, що зазвичай важко зустріти на практиці. З огляду на це автором запропонована власна модель динамічного прайсингу в дискретному часі з періодичним коригуванням цінової політики. Модель враховує також параметричну («ідеальну») криву продажів і функцію розподілу купівельних цін. Ці характеристики оцінюються за допомогою історичних даних. Завдання моделі - запропонувати оптимальну ціну пропозиції, яка б дозволяла продавати товар згідно темпу параметричної кривої і максимізувала прибуток підприємства. Запропонована методика була успішно протестована для випадку продажу туристичних турів. Подальша розробка моделі передбачає використання байєсівських мереж для калібрування оцінок її параметрів.
... For the deterministic programming, we can construct effective algorithm to solve. Kleywegt (2001) constructed a pricing and overbooking model in continuous time, and he used Lagrangian duality to solve the model. Topaloglu (2009) showed that decomposition methods could be visualized as an application of Lagrangian relaxation to the dynamic programming formulation of the network revenue management problem. ...
In this paper, based on the single-leg revenue management, we develop a network revenue management model to jointly make the capacity allocation and overbooking decisions. In our model, the factors such as no-show, late cancellation, and denied boarding are taken into account. Both no-show and late cancellation have uncertainties. Therefore, we make use of a stochastic chance constrained integer programming to formulate the capacity allocation and overbooking problem. The model can be converted to determined integer programming to be solved by Monte Carlo algorithm. At last, in order to test the validity and feasibility of the model, we conduct computational experiments. We demonstrate that the model that is able to effectively avoid the economic loss by no-show, late cancellation or denied boarding, while keeping the expected revenue of airlines stable.
... He also provides an upper bound for the revenue generated, based on a deterministic mathematical program. Kleywegt (2001) [19] provides an optimal control for the dynamic pricing problem. He also considers the two constraints, cancellation, and refund and assumes they are independent. ...
In this paper we simultaneously address four constraints relevant to airline revenue management problem: flight cancellation, customer no-shows, overbooking, and refunding. We develop a linear program closely related to the dynamic program formulation of the problem, which we later use to approximate the optimal decision rule for rejecting or accepting customers. First, we give a novel proof that the optimal objective function of this linear program is always an upper bound for the dynamic program. Secondly, we construct a decision rule based on this linear program and prove that it is asymptotically optimal under certain circumstances. Finally, using Monte Carlo simulation, we demonstrate that, numerically, the result of the linear programming policy presented in this paper has a short distance to the upper bound of the optimal answer, which makes it a fairly good approximate answer to the intractable dynamic program. © 2019 Faculty of Organizational Sciences, Belgrade. All Rights Reserved.
... Another important paper related to ours is Kleywegt (2001), where the author considers a stylized (deterministic) fluid model of a general dynamic pricing problem for selling a network of resources. In Kleywegt's model prices are chosen dynamically to sell products (or itineraries) to multiple customer classes over time. ...
We consider a network revenue management problem and advance its dual formulation. The dual formulation reveals that the (optimal) shadow price of capacity forms a nonnegative martingale. This result is proved under minimal assumptions on network topology and stochastic nature of demand, allowing an arbitrary statistical dependence structure across time and products. Next, we consider a quadratic perturbation of the network revenue management problem and show that a simple (perturbed) bid-price control is optimal for the perturbed problem; and it is ɛ-optimal for the original network revenue management problem. Finally, we consider a predictable version of this control, where the bid prices used in the current period are last updated in the previous period, and provide an upper bound on its optimality gap in terms of the (quadratic) variation of demand. Using this upper bound we show that there exists a near-optimal such control in the usual case when periods are small compared to the planning horizon provided that either demand or the incremental information arriving during each period is small. We establish the martingale property of the (near) optimal bid prices in both settings. The martingale property can have important implications in practice as it may offer a tool for monitoring the revenue management systems.
... Karaesmen and van Ryzin (2004b) describe a capacity allocation and overbooking model that is useful when dealing with multiple flight legs that can serve as substitutes of each other. Kleywegt (2001) develops a joint pricing and overbooking model over an airline network assuming that the reservation requests are deterministic and he solves it using duality and decomposition ideas. Finally, the book Talluri and van Ryzin (2004) contains background and details on revenue management, specifically the chapters on overbooking and network revenue management. ...
... Several improvements to these were introduced later, through models of varying complexity, that strove to closely replicate different real-life situations, such as multiple product scenarios (see [25,9,30]). A Markov decision process in a heuristic [3] and deterministic [20] sense has also been used to solve the pricing problem. ...
... This work has led to a number of extensions that investigate the dynamic pricing policies of monopolists. Most notably, Gallego and van Ryzin (1997) and Kleywegt (2001) Bitran and Caldenty (2002) and Elmaghraby and Keskinocak (2003) present extensive reviews of dynamic pricing models in revenue management. In all of these models, firms' optimal price policies are dynamic either due to demand uncertainty or due to timeinhomogeneous demand process. ...
This paper presents a two-period model to analyze the dynamic pricing behavior of two profit-maximizing firms that have equal inventories of perfectly substitutable and perishable products. Two versions of the problem are studied. In the first version, firms simultaneously announce their first period prices, observe the sales each makes, and then simultaneously an-nounce their second period prices. In the second version, firms may restrict their sales in the first period by allocating a portion of their inventories. In each period, customers first purchase from the low price firm and then from the high price firm up to their inventories, provided that the prices are lower than their reservation prices. It is shown that a dynamic pricing behavior exists even when there is no uncertainty and no exogenous temporal variation in consumer characteristics. Secondly, it is observed that inventory control may have positive as well as adverse effects on firm profits. Finally, firms are not able to achieve a collusive outcome even when the total inventory is less than the potential demand.
... We use a variant of their approach as a benchmark strategy in our computational experiments. Kleywegt (2001) develops a joint pricing and overbooking model, where the itinerary requests are deterministic functions of the prices and he solves the model by using Lagrangian duality arguments. Zhang and Cooper (2009) consider the problem of pricing substitutable flights that operate between the same origin destination pair. ...
In this paper, we develop two methods for making pricing decisions in network revenue management problems. We consider a setting where the probability of observing a request for an itinerary depends on the prices and the objective is to dynamically adjust the prices so as to maximize the total expected revenue. The idea behind both of our methods is to decompose the dynamic programming formulation of the pricing problem by the ∞ight legs and to obtain value function approximations by focusing on one ∞ight leg at a time. We show that our methods provide upper bounds on the optimal total expected revenue and these upper bounds are tighter than the one provided by a deterministic linear program commonly used in practice. Our computational experiments yield two important results. First, our methods provide substantial improvements over the deterministic linear program. The average gap between the total expected revenues obtained by our methods and the deterministic linear program is 7.11%. On average, our methods tighten the upper bounds obtained by the deterministic linear program by 3.66%. Second, the two methods that we develop have difierent strengths. In particular, while one method is able to obtain tighter upper bounds, the other one is able to obtain pricing policies that yield higher total expected revenues.
... We restrict ourselves to such quantity-based techniques since price-based techniques quickly result in far more difficult models for the problem that we consider. For models for a dynamic pricing approach to airline revenue management, we refer to: Chatwin (2000), Gallego (1995, 2000), Feng and Xiao (2000a, b), van Ryzin (1994, 1997), Kleywegt (2001), Maglaras and Meissner (2004), Walczak (2003), You (1999) and Zhao and Zheng (2000) among others. See also Bitran and Caldentey (2003) who provide an overview of pricing models for revenue management problems. ...
Kevin Pak (1977) obtained his Master’s degree in Econometrics and
Operations Research from the Erasmus University Rotterdam in 2000. In the same year he joined ERIM in order to carry out his doctoral research on the subject of revenue management. Throughout the years his work has been published in and presented at a number of international journals and conferences. Currently, he applies his knowledge of operations research into practice as a consultant at ORTEC bv.
... For the majority of work in this literature, there is a one time perishable capacity. Kleywegt (2001) considers dynamic pricing of multiple products to multiple customer classes over time, where the products share a number of scarce resources. Maglaras and Meissner (2006) model dynamic pricing in the presence of a …xed and …nite capacity. ...
We consider a make-to-order …rm that has the ability to dynamically o¤er menus of prices and production (or service) leadtimes to its customers. Customers seeking a particular product (or service) will choose from the o¤ered menu the pair of prices and leadtimes that maximizes their value for the product minus their delay cost and the price for that leadtime. Dynamic control allows the retailer to tune leadtimes and prices to the current backlog. We consider two classes of customers who have the same valuation for the product but di¤er in their level of patience (a concept made precise in the paper). We investigate how such dynamic menus should be chosen in the context of a large capacity asymptotic regime and propose policies when customers'leadtime costs are convex-concave. We consider both the full information case and the (more realistic) case where the …rm, being unaware of customer type, must o¤er incentive-compatible menus. We propose readily-implementable policies and test them numerically against a number of natural benchmarks.
... ight leg capacity constraints are usually used as proxies for the expected marginal values of each unit of capacity. These dual variables are used to control sales through a bid price policy. Erdelyi and Topaloglu (2009) show that the optimal objective value of the deterministic linear program is an upper bound on the optimal expected total profit. Kleywegt (2001) develops a joint pricing and overbooking model over an airline network assuming that the reservation requests are deterministic. He solves the model by using duality and decomposition ideas. Karaesmen and van Ryzin (2004b) describe a capacity allocation and overbooking model that is useful when dealing with multiple flight legs that can ...
Revenue management practices often include overbooking capacity to account for customers who make reservations but do not show up. In this paper, we consider the network revenue management problem with no-shows and overbooking, where the show-up probabilities are specific to each product. No-show rates differ significantly by product (for instance, each itinerary and fare combination for an airline) as sale restrictions and the demand characteristics vary by product. However, models that consider no-show rates by each individual product are difficult to handle as the state-space in dynamic programming formulations (or the variable space in approximations) increases significantly. In this paper, we propose a randomized linear program to jointly make the capacity control and overbooking decisions with product-specific no-shows. We establish that our formulation gives an upper bound on the optimal expected total profit and our upper bound is tighter than a deterministic linear programming upper bound that appears in the existing literature. Furthermore, we show that our upper bound is asymptotically tight in a regime where the leg capacities and the expected demand is scaled linearly with the same rate. We also describe how the randomized linear program can be used to obtain a bid price control policy. Computational experiments indicate that our approach is quite fast, able to scale to industrial problems and can provide significant improvements over standard benchmarks.
... This reduction could prove computationally beneficial, since as is often the case the number of products (e.g., the number of fare-class and origin-destination pairs) tends to be greater than the number of resources (e.g., number of flights in a hub-and-spoke network). We refer the reader to Gallego and van Ryzin [10] and Kleywegt [14] for fluid formulations to multi-product network revenue management problems. ...
This chapter reviews multi-product dynamic pricing models for a revenue maximizing mo-nopolist firm. The baseline model studied in this chapter is of a seller that owns a fixed capacity of a resource that is consumed in the production or delivery of some type of product. The seller selects a dynamic pricing strategy for the offered product so as to maximize its total expected revenues over a finite time horizon. We then review how this model can be extended to settings where the firm is selling multiple products that consume this firm's capacity, and finally highlight a connection between these dynamic pricing models and the closely related model where prices are fixed, and the seller dynamically controls how to allocate capacity to requests for the different products. Methodologically, this chapter reviews the dynamic programming formulations of the above problems, as well as their associated deterministic (fluid) analogues. It highlights some of the key insights and pricing heuristics that are known for these problems, and briefly mentions possible extensions and areas of current interest.
... Luo [42] considers a make-to-stock multiclass queueing scheduling problem that minimizes a convex quadratic backorder and holding cost and finds an optimal production policy over the entire time horizon. Kleywegt [38] uses a cutting plane algorithm to solve a multiclass optimal control problem of dynamic pricing with profit linear in terms of selling rate. Fleisher and Sethuraman [26] provide an approximation algorithm to solve the optimal control of fluid queueing networks. ...
In this paper, we present a continuous time optimal control model for studying a dynamic pricing and inventory control problem for a make-to-stock manufacturing system. We consider a multiproduct capacitated, dynamic setting. We introduce a demand-based model where the demand is a linear function of the price, the inventory cost is linear, the production cost is an increasing strictly convex function of the production rate, and all coefficients are time-dependent. A key part of the model is that no backorders are allowed. We introduce and study an algorithm that computes the optimal production and pricing policy as a function of the time on a finite time horizon, and discuss some insights. Our results illustrate the role of capacity and the effects of the dynamic nature of demand in the model. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007
... Karaesmen and van Ryzin (2004b) describe a capacity allocation and overbooking model that is useful when dealing with multiple flight legs that can serve as substitutes of each other, which is the case for multiple flights in a day that connect the same origin destination pair. Kleywegt (2001) develops a joint pricing and overbooking model over an airline network. This model assumes that the itinerary requests are deterministic and it is solved by using decomposition and duality ideas. ...
In this paper, we develop a revenue management model to jointly make the capacity allocation and overbooking decisions over
an airline network. The crucial observation behind our model is that if the penalty cost of denying boarding to the reservations
were given by a separable function, then the optimality equation for the joint capacity allocation and overbooking problem
would decompose by the itineraries. We exploit this observation by building an approximation to the penalty cost that is separable
by the numbers of reservations for different itineraries. In this case, we can obtain an approximate solution to the optimality
equation by plugging the separable approximation into the boundary condition of the optimality equation. Our computational
experiments compare our approach with a standard deterministic linear programming formulation, as well as a recent joint capacity
allocation and overbooking model. When compared with the standard deterministic linear programming formulation, our approach
can provide significant profit improvements. On the other hand, when compared with the recent joint capacity allocation and
overbooking model, our approach can provide similar profit performance with substantially shorter runtimes.
... In addition, [24], [25] present a broad overview of pricing and revenue management issues and provide a unified modeling framework focused on dynamic pricing models in revenue management. [26], [27], [28] formulate complex stochastic dynamic programming models for the pricing problem, where demand is modeled as a stochastic process with price-dependent intensity. These works provide structural results and heuristics based on the deterministic versions of the pricing problem. ...
In this paper, we analyze a round-based pricing scheme that encourages favorable behavior from users of real-time P2P applications like P2PTV. In the design of pricing schemes, we consider price to be a function of usage and capacity of download/upload streams, and quality of content served. Users are consumers and servers at the same time in such networks, and often exhibit behavior that is unfavorable towards maximization of social benefits. Traditionally, network designers have overcome this difficulty by building-in traffic latencies. However, using simulations, we show that appropriate pricing schemes and usage terms can enable designers to limit required traffic latencies, and be able to earn nearly 30% extra revenue from providing P2PTV services. The service provider adjusts the prices of individual programs incrementally within rounds, while making relatively large-scale adjustments at the end of each round. Through simulations, we show that it is most beneficial for the service provider to carry out 5 such rounds of price adjustments for maximizing his average profit and minimizing the associated standard deviation at the same time.
... ue from the itinerary requests. Gallego and van Ryzin (1997) provide theoretical support for the deterministic linear program by showing that the control policy obtained from a variant of the deterministic linear program is asymptotically optimal as the leg capacities and the expected number of itinerary requests increase linearly at the same rate. Kleywegt (2001) constructs a pricing and overbooking model in continuous time. The demand process that he uses is deterministic and he utilizes Lagrangian duality to solve the model. The literature on decomposition of network revenue management problems is also related to our paper. Williamson (1992) is one of the first to decompose the network revenue ...
In this paper, we develop a revenue management model to jointly make the capacity allocation and overbooking decisions over an airline network. Our approach begins with the dynamic programming formulation of the capacity allocation and overbooking problem and uses an approximation strategy to decompose the dynamic programming formulation by the ∞ight legs. This decomposition idea opens up the possibility of obtaining approximate solutions by concentrating on one ∞ight leg at a time, but the capacity allocation and overbooking problem that takes place over a single ∞ight leg still turns out to be intractable. We use a state aggregation approach to obtain high quality solutions to the single leg problem. Overall, our model constructs separable approximations to the value functions, which can be used to make the capacity allocation and overbooking decisions for the whole airline network. Computational experiments indicate that our model performs signiflcantly better than a variety of benchmark strategies from the literature.
... In part, this paper extends their results by adding a control dimension to some of their models. The use of fluid models for dynamic pricing in manufacturing systems has been discussed in Kleywegt (2001), while the literature on fluid models for purposes of sequencing and routing control is large; see, e.g., Chen and Yao (1993), Avram et al. (1995), Maglaras (2000, and the references therein. Workload formulations arise in stochastic network control problems that are considered in Operations Research 54(5), pp. ...
Motivated by the recent adoption of tactical pricing strategies in manufacturing settings, this paper studies a problem of dynamic pricing for a multiproduct make-to-order system. Specifically, for a multiclass Mn/M/1 queue with controllable arrival rates, general demand curves, and linear holding costs, we study the problem of maximizing the expected revenues minus holding costs by selecting a pair of dynamic pricing and sequencing policies. Using a deterministic and continuous (fluid model) relaxation of this problem, which can be justified asymptotically as the capacity and the potential demand grow large, we show the following: (i) greedy sequencing (i.e., the cμ-rule) is optimal, (ii) the optimal pricing and sequencing decisions decouple in finite time, after which (iii) the system evolution and thus the optimal prices depend only on the total workload. Building on (i)--(iii), we propose a one-dimensional workload relaxation to the fluid pricing problem that is simpler to analyze, and leads to intuitive and implementable pricing heuristics. Numerical results illustrate the near-optimal performance of the fluid heuristics and the benefits from dynamic pricing.
... We use the optimization problem proposed by Gallego and van Ryzin (1997) as a benchmark strategy. Kleywegt (2001) develops a joint pricing and overbooking model over an airline network. He assumes that the demands for the itineraries are deterministic functions of the prices and solves the model by using Lagrangian duality ideas. ...
In this paper, we develop a stochastic approximation algorithm for making pricing decisions in network revenue management problems. In the setting we consider, the probability of observing a request for an itinerary depends on the price for the itinerary. We are interested in flnding a set of prices that maximize the total expected revenue. Our approach is based on visualizing the total expected revenue as a function of the prices and using the stochastic gradients of the total revenue to search for a good set of prices. To compute the stochastic gradients of the total revenue, we use a novel construction that decouples the prices for the itineraries from the probability distributions of the itinerary requests. This construction ensures that the probability distributions of the underlying random variables do not change when we change the prices for the itineraries. We establish the convergence of our stochastic approximation algorithm. Computational experiments indicate that the prices obtained by our stochastic approximation algorithm perform signiflcantly better than those obtained by standard benchmark strategies, especially when the leg capacities are tight and there are large difierences between the price sensitivities of the difierent market segments.
... This work has led to a number of extensions. Most notably, Gallego and van Ryzin (1997) and Kleywegt (2001) Ryzin in several directions. They have presented game-theoretic models that include product variety and price decisions in addition to the inventory decisions. ...
This paper analyzes the impact of dynamic and fixed-ratio pricing policies on firm profits and equilibrium prices under competition. Firms that have equal inventories of perfectly substitutable and perishable products compete for customer segments that demand the product at different times. In each period, customers first purchase from the low price firm and then from the high price firm up to their inventories, provided the prices are lower than the maximum they are willing to pay. The main conclusions of this paper are as follows: although dynamic pricing is a more sophisticated policy than fixed-ratio pricing, it may lead to decreased equilibrium profits; under both pricing policies, one firm assumes the role of a low-cost high-output firm while the other assumes the role of a high-cost low-output firm; and, the supply demand ratio has more impact on the outcome of the competition than the heterogeneity in consumer reservation prices.
... In this paper, we will, however, not consider this situation. Applications of pricing techniques to airline revenue management can be found in Chatwin (2000), Gallego (1995, 2000), Xiao (2000a, 2000b), van Ryzin (1994, 1997), Kleywegt (2001), You (1999) and Zhao and Zheng (2000) among others. ...
With the increasing interest in decision support systems and the continuous advance of computer science, revenue management is a discipline which has received a great deal of interest in recent years. Although revenue management has seen many new applications throughout the years, the main focus of research continues to be the airline industry. Ever since Littlewood (1972) first proposed a solution method for the airline revenue management problem, a variety of solution methods have been introduced. In this paper we will give an overview of the solution methods presented throughout the literature.
... In addition, [24], [25] present a broad overview of pricing and revenue management issues and provide a unified modeling framework focused on dynamic pricing models in revenue management. [26], [27], [28] formulate complex stochastic dynamic programming models for the pricing problem, where demand is modeled as a stochastic process with price-dependent intensity. These works provide structural results and heuristics based on the deterministic versions of the pricing problem. ...
In this paper, we analyze a round-based pricing scheme that encourages favorable behavior from users of real-time P2P applications like P2PTV. In the design of pricing schemes, we consider price to be a function of usage and capacity of download/upload streams, and quality of content served. Users are consumers and servers at the same time in such networks, and often exhibit behavior that is unfavorable towards maximization of social benefits. Traditionally, network designers have overcome this difficulty by building-in traffic latencies. However, using simulations, we show that appropriate pricing schemes and usage terms can enable designers to limit required traffic latencies, and be able to earn nearly 30% extra revenue from providing P2PTV services. The service provider adjusts the prices of individual programs incrementally within rounds, while making relatively large-scale adjustments at the end of each round. Through simulations, we show that it is most beneficial for the service provider to carry out 5 such rounds of price adjustments for maximizing his average profit and minimizing the associated standard deviation at the same time.
... Gallego and van Ryzin [25] and Paschalidis and Tsitsiklis [35] extend this type of model to the dynamic pricing of multiple products whose production draws from a shared supply of resources. Kleywegt [30] gives an optimal control formulation of the multi-period dynamic pricing problem. Kachani and Perakis [28] propose a deterministic fluid model for dynamic pricing and inventory management for non-perishable products in capacitated and competitive make-to-stock manufacturing systems. ...
This paper studies the problem of multi-period pricing for perishable products in a competitive (oligopolistic) market. We study non cooperative Nash equilibrium policies for sellers. At the beginning of the time horizon, the total inventories are given and additional production is not an available option. The analysis for periodic production-review models, where production decisions can be made at the end of each period at some production cost after incurring holding or backorder costs, does not extend to this model. Using results from game theory and variational inequalities we study the existence and uniqueness of equilibrium policies. We also study convergence results for an algorithm that computes the equilibrium policies. The model in this paper can be used in a number of application areas including the airline, service and retail industries. We illustrate our results through some numerical examples. Singapore-MIT Alliance (SMA)
... In this paper, we will, however, not consider this situation. Applications of pricing techniques to airline revenue management can be found in Chatwin (2000), Gallego (1995, 2000), Xiao (2000a, 2000b), van Ryzin (1994, 1997), Kleywegt (2001), You (1999) and Zhao and Zheng (2000) among others. ...
With the increasing interest in decision support systems and the continuous advance of computer science, revenue management is a discipline which has received a great deal of interest in recent years. Although revenue management has seen many new applications throughout the years, the main focus of research continues to be the airline industry. Ever since Littlewood (1972) first proposed a solution method for the airline revenue management problem, a variety of solution methods have been introduced. In this paper we will give an overview of the solution methods presented throughout the literature.
E-commerce in the present age has led to a review of all strategies in business and commerce. One of these changes is changes in the pricing policies and strategies of goods and services. Due to the intensification of competition in the business environment, especially electronic businesses, fixed and static pricing can not meet the needs of sellers who offer their products in this market. Therefore, these sellers are looking for methods based on dynamic pricing so that they can change the price of their products in these turbulent markets and react appropriately to this competitive environment. The main purpose of this study is to investigate and analyze the factors affecting consumer behavior in online shopping in online stores in Mashhad. From the perspective of purpose, this research is of applied development type and from the methodological point of view is of descriptive research from the branch of correlation. The results show that the factors of purchase price of product, quality of goods and services, reputation of the company brand, after-sales service, method and time of delivery of products and offering discounts were the most important factors that affect consumer behavior in online shopping and these factors With high accuracy and predictive power, they predict 93.33% of consumer behavior in online shopping.
Early on, many airlines adopted the policy of not penalizing booked customers for canceling reservations at any time before departure. Some would not even penalize those that did not show up for booked flights. In essence, an airline ticket was “like money” since it could be used at full face value for a future flight or redeemed for cash at any future date. In the sixties, no-shows were becoming a problem for airlines who found that flights that were fully booked were departing with many empty seats. In response, the airlines began to overbook as a means of hedging against no-shows. If a flight had more passengers show than there were seats available, then the airlines would bump some passengers. The bumped passengers would be re-booked on a later flight. In addition, bumped passengers would be given other compensation, often a meal at the airport and a discount certificate applicable to future travel. The cost to the airline of bumping a passenger is called the denied boarding cost. The denied boarding cost would include the cost of putting a bumped passenger on another flight to her destination, the cost of any direct compensation to the bumped passenger, the cost of the meals or lodging that the airline provides to each bumped passenger, and the cost of “ill will” incurred by bumping the passenger. These costs can be different for each flight. For example, a passenger bumped from the last flight of the day will be provided with a hotel room at the airline’s expense.
The topic of dynamic pricing and learning has received a considerable amount of attention in recent years, from different scientific communities. We survey these literature streams: we provide a brief introduction to the historical origins of quantitative research on pricing and demand estimation, point to different subfields in the area of dynamic pricing, and provide an in-depth overview of the available literature on dynamic pricing and learning. Our focus is on the operations research and management science literature, but we also discuss relevant contributions from marketing, economics, econometrics, and computer science. We discuss relations with methodologically related research areas, and identify directions for future research.
We develop a network revenue management model to jointly make capacity control and overbooking decisions. Our approach is based on the observation that if the penalty cost of denying boarding to the reservations at the departure time were given by a separable function, then the dynamic programming formulation of the network revenue management problem would decompose by the itineraries and it could be solved by focusing on one itinerary at a time. Motivated by this observation, we use an iterative and simulation-based method to build separable approximations to the penalty cost that we incur at the departure time. Computational experiments compare our model with two benchmark strategies that are based on a deterministic linear programming formulation. The profits obtained by our model improve over those obtained by the benchmark strategies by about 3% on the average, which is a significant figure in the network revenue management setting. For the test problems with tight leg capacities, the profit improvements can be as high as 13%.
In this article, we develop a stochastic approximation algorithm to find good bid price policies for the joint capacity allocation and overbooking problem over an airline network. Our approach is based on visualizing the total expected profit as a function of the bid prices and searching for a good set of bid prices by using the stochastic gradients of the total expected profit function. We show that the total expected profit function that we use is differentiable with respect to the bid prices and derive a simple expression that can be used to compute its stochastic gradients. We show that the iterates of our stochastic approximation algorithm converge to a stationary point of the total expected profit function with probability 1. Our computational experiments indicate that the bid prices computed by our approach perform significantly better than those computed by standard benchmark strategies and the performance of our approach is relatively insensitive to the frequency with which we recompute the bid prices over the planning horizon. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011
We develop a Markov decision process formulation of a dynamic pricing problem for multiple substitutable flights between the same origin and destination, taking into account customer choice among the flights. The model is rendered computationally intractable for exact solution by its multi-dimensional state and action spaces, so we develop and analyze various bounds and heuristics. We first describe three related models, each based on some form of pooling, and introduce heuristics suggested by these models. We also develop separable bounds for the value function which are used to construct value- and policy-approximation heuristics. Extensive numerical experiments show the value- and policy-approximation approaches to work well across a wide range of problem parameters, and to outperform the pooling-based heuristics in most cases. The methods are applicable even for large problems, and are potentially useful for practical applications.
We consider a revenue maximizing make-to-order manufacturer that serves a market of price and delay sensitive customers and operates in an environment in which the market size varies stochastically over time. A key feature of our analysis is that no model is assumed for the evolution of the market size. We analyze two main settings: i) the size of the market is observable at any point in time; and ii) the size of the market is not observable and hence cannot be used for decision-making. We focus on high-volume systems that are characterized by large processing capacities and market sizes, and where the latter fluctuate on a slower time scale than that of the underlying production system dynamics. We develop an approach to tackle such problems that is based on an asymptotic analysis and that yields near-optimal policy recommendations for the original system via the solution of a stochastic fluid model.
This paper examines a dynamic general equilibrium model with supply friction. With or without friction, the competitive equilibrium is efficient. Without friction, the market price is completely determined by the marginal production cost. If friction is present, no matter how small, then the market price fluctuates between zero and the "choke-up" price, without any tendency to converge to the marginal production cost, exhibiting considerable volatility. The distribution of the gains from trading in an efficient allocation may be skewed in favor of the supplier, although every player in the market is a price taker.
The concept of yield management for IT infrastructures, and in particular for On Demand IT utilities was recently introduced in . IT On Demand is a business model touted recently by IBM and other large IT companies as the future of computing. In , it was demonstrated that this paradigm is appropriate for, and can benefit greatly from, use of a convenient yield management approach, and an optimization model and theoretical justification of it were presented. The present paper provides a detailed analysis of that model, both in simplified cases where an analytical analysis is possible, and numerically on larger problem instances, and confirms the significant revenue benefit that can accrue through use of yield management in the IT On Demand context.
Recently, revenue management has become popular in many industries such as the airline, the supply chain, and the transportation industry. Decision makers realize that even small improvements in their operations can have a significant impact on their profits. Nevertheless, determining pricing and inventory optimal policies in more realistic settings may not be a tractable task. Ignoring the potential inaccuracy of parameters may lead to a solution that actually performs poorly, or even that violates some constraints. Finally, competitors impact a supplier's best strategy by influencing her demand, revenues, and field of possible actions. Taking a game theoretic approach and determining the equilibrium of the system can help understand its state in the long run. This thesis presents a continuous time optimal control model for studying a dynamic pricing and inventory control problem in a make-to-stock manufacturing system. We consider a multi-product capacitated, dynamic setting. We introduce a demand-based model with convex costs. A key part of the model is that no backorders are allowed, as this introduces a constraint on the state variables. We first study the deterministic version of this problem.
We present an optimization approach for jointly learning the demand as a functionof price, and dynamically setting prices of products in an oligopoly environment in order to maximize expected revenue. The models we consider do not assume that the demand as a function of price is known in advance, but rather assume parametric families of demand functions that are learned over time. We first consider the noncompetitive case and present dynamic programming algorithms of increasing computational intensity with incomplete state information for jointly estimating the demand and setting prices as time evolves. Our computational results suggest that dynamic programming based methods outperform myopic policies often significantly. We then extend our analysis in a competitive environment with two firms. We introduce a more sophisticated model of demand learning, in which the price elasticities are slowly varying functions of time, and allows for increased flexibility in the modeling of the demand. We propose methods based on optimization for jointly estimating the Firm's own demand, its competitor's demand, and setting prices. In preliminary computational work, we found that optimization based pricing methods offer increased expected revenue for a firm independently of the policy the competitor firm is following.
Thesis (Ph. D.)--Columbia University, 2004. Includes bibliographical references (leaves 157-167). Department: Graduate School of Business.
With the increasing interest in decision support systems and the continuous advance of computer science, revenue management is a discipline which has received a great deal of interest in recent years. Although revenue management has seen many new applications throughout the years, the main focus of research continues to be the airline industry. Ever since LITTLEWOOD (1972) first proposed a solution method for the airline revenue management problem, a variety of solution methods have been introduced. In this paper we give an overview of the solution methods presented throughout the literature.
Yield management for e-services is an exciting extension of the traditional yield management techniques employed in hotel and airline industry. We propose a framework for realtime yield management of e-services. The framework employs demand forecasting, yield management and customer behavior monitoring modules. Yield management is used to optimize resource allocation with market segmentation in prices. In e-services an important service measure is the perceived delay. We use queueing theory to model the dependence of the perceived delay on the demand and providers' capacity. Using numerical examples we show the gain in revenue that can be realised by firms in different types of markets with different buying power of customers
This survey reviews the forty-year history of research on transportation revenue management (also known as yield management). We cover developments in forecasting, overbooking, seat inventory control, and pricing, as they relate to revenue management, and suggest future research directions. The survey includes a glossary of revenue management terminology and a bibliography of over 190 references. In the forty years since the first publication on overbooking control, passenger reservations systems have evolved from low level inventory control processes to major strategic information systems. Today, airlines and other transportation companies view revenue management systems and related information technologies as critical determinants of future success. Indeed, expectations of revenue gains that are possible with expanded revenue management capabilities are now driving the acquisition
In this chapter we study parameterized variational inequalities (generalized equations) and discuss applications of the theory to nonlinear, semi-definite and semi-infinite programming problems. Various aspects of these specific applications of the general theory have been discussed in the previous chapters. However, we recommand to readers primarily interested in one of these applications to read the corresponding section of this chapter first, since it allows one to have a global view of the power of optimality conditions and perturbation theory for these topics. In addition, this chapter provides some results that were not presented in the previous chapters.
This book may be regarded as consisting of two parts. In Chapters I-IV we pre sent what we regard as essential topics in an introduction to deterministic optimal control theory. This material has been used by the authors for one semester graduate-level courses at Brown University and the University of Kentucky. The simplest problem in calculus of variations is taken as the point of departure, in Chapter I. Chapters II, III, and IV deal with necessary conditions for an opti mum, existence and regularity theorems for optimal controls, and the method of dynamic programming. The beginning reader may find it useful first to learn the main results, corollaries, and examples. These tend to be found in the earlier parts of each chapter. We have deliberately postponed some difficult technical proofs to later parts of these chapters. In the second part of the book we give an introduction to stochastic optimal control for Markov diffusion processes. Our treatment follows the dynamic pro gramming method, and depends on the intimate relationship between second order partial differential equations of parabolic type and stochastic differential equations. This relationship is reviewed in Chapter V, which may be read inde pendently of Chapters I-IV. Chapter VI is based to a considerable extent on the authors' work in stochastic control since 1961. It also includes two other topics important for applications, namely, the solution to the stochastic linear regulator and the separation principle.
This chapter discusses the method of multipliers for equality constrained problems. By solving an approximate problem, an approximate solution of the original problem can be obtained. However, if a sequence of approximate problems can be constructed that converges in a well-defined sense to the original problem, then the corresponding sequence of approximate solutions would yield in the limit a solution of the original problem. The basic idea in penalty methods is to eliminate some or all of the constraints and add to the objective function a penalty term that prescribes a high cost to infeasible points. A parameter that determines the severity of the penalty and as a consequence the extent to which the resulting unconstrained problem approximates the original constrained problem is associated with the penalty methods.
In 1993, National Car Rental faced liquidation. General Motors Corporation (National's parent) took a 56 million in the first year. In April 1995, General Motors sold National Car Rental Systems for an estimated $1.2 billion.
The main subject of this book is perturbation analysis of continuous optimization problems. In the last two decades considerable progress has been made in that area, and it seems that it is time now to present a synthetic view of many important results that apply to various classes of problems.
A problem of minimizing a sum of many convex piecewise-linear functions is considered. In view of applications to two-stage linear programming, where objectives are marginal values of lower level problems, it is assumed that domains of objectives may be proper polyhedral subsets of the space of decision variables and are defined by piecewise-linear induced feasibility constraints. We propose a new decomposition method that may start from an arbitrary point and simultaneously processes objective and feasibility cuts for each component. The master program is augmented with a quadratic regularizing term and comprises an a priori bounded number of cuts. The method goes through nonbasic points, in general, and is finitely convergent without any nondegeneracy assumptions. Next, we present a special technique for solving the regularized master problem that uses an active set strategy and QR factorization and exploits the structure of the master. Finally, some numerical evidence is given.
Proximal bundle methods for minimizing a convex functionf generate a sequence {xk} by takingxk+1 to be the minimizer of
, where
is a sufficiently accurate polyhedral approximation tof anduk > 0. The usual choice ofuk = 1 may yield very slow convergence. A technique is given for choosing {uk} adaptively that eliminates sensitivity to objective scaling. Some encouraging numerical experience is reported.
Recent Developments in Algorithms and Software for Trust Region Methods. In Mathematical Programming, the State of the
- J J Moré
Moré, J. J. 1997. Recent Developments in Algorithms and Software for Trust Region Methods.
In Mathematical Programming, the State of the Art. A. Bachem, M. Grötschel and B. Korte
(editors). Springer Verlag, Berlin, Germany, 258-287.
- B C Smith
- J F Leimkuhler
- R M Darrow
Smith, B. C., Leimkuhler, J. F. and Darrow, R. M. 1992. Yield Management at American
Airlines. Interfaces, 22, 8-31.