Dynamic Pricing Strategies for Multi-Product …

1 MANUFACTURING&SERVICEOPERATIONSMANAGEMEN TVol. 8, No. 2, Spring 2006, pp. 136 148issn1523-4614 eissn1526-5498 06 0802 0136informs 2006 INFORMSD ynamic Pricing Strategies for MultiproductRevenue Management ProblemsConstantinos MaglarasColumbia Business School, Columbia University, 409 Uris Hall, 3022 Broadway,New York, New York 10027, MeissnerLancaster University Management School, Lancaster University, Room A48,Lancaster LA1 4XY, United Kingdom, a firm that owns a fixed capacity of a resource that is consumed in the production or delivery ofmultiple products. The firm strives to maximize its total expected revenues over a finite horizon, eitherby choosing a Dynamic Pricing strategy for each product or, if prices are fixed, by selecting a Dynamic rulethat controls the allocation of capacity to requests for the different products.

2 This paper shows how these well-studied revenue management problems can be reduced to a common formulation in which the firm controlsthe aggregate rate at which all products jointly consume resource capacity, highlighting their common structure,and in some cases leading to algorithmic simplifications through the reduction in the control dimension ofthe associated optimization problems. In the context of their associated deterministic (fluid) formulations, thisreduction leads to a closed-form characterization of the optimal controls, and suggests several natural static anddynamic Pricing heuristics. These are analyzed asymptotically and through an extensive numerical study. In thecontext of the former, we show that resolving the fluid heuristic achieves asymptotically optimal performanceunder fluid words: revenue management; Dynamic Pricing ; capacity controls; fluid approximations; efficient frontierHistory: Received: July 21, 2003; accepted: November 11, 2005.

3 This paper was with the authors 9 months for3 IntroductionConsider a firm that owns a fixed capacity of a cer-tain resource that is consumed in the process of pro-ducing or offering multiple products or services, andwhich must be consumed over a finite time firm s problem is to maximize its total expectedrevenues by selecting the appropriate Dynamic con-trols. We consider two well-studied variants of thisproblem. In the first, the firm is assumed to be amonopolist or to operate in a market with imper-fect competition, and thus to have power to influ-ence the demand for each product by varying itsprice. In this setting, the firm s problem is to choosea Dynamic Pricing strategy for each of its productsto optimize expected revenues. In the second variant,prices are assumed to be fixed either by the competi-tion or through a higher-order optimization problem,and the firm s problem is now to choose a dynamiccapacity allocation rule that controls when to acceptnew requests for each of these products.

4 In the sequel,these two problems are referred to as the dynamicpricing and capacity control formulations, respec-tively. Revenue management problems of that sortgained interest in the late 1970s in the context ofthe airline industry, and have since been successfullyintroduced in a variety of other areas such as hotels,cruise lines, rental cars, retail, paper illustrates how these two problems canbe reduced to a common formulation, thus connect-ing prior results that have appeared in the literatureunder a unified framework, and explores some ofthe consequences of this formulation. Specifically, weshow that the multiproduct Dynamic Pricing prob-lem introduced by Gallego and van Ryzin (1997) andthe capacity control problem of Lee and Hersh (1993)can be recast within this common framework, andbe treated as different instances of a single-productpricing problem for appropriate concave revenue136 Maglaras and Meissner: Dynamic Pricing Strategies for Multiproduct Revenue Management ProblemsManufacturing & Service Operations Management 8(2), pp.

5 136 148, 2006 INFORMS137functions (Propositions 1 and 2). Broadly speaking,this is done by decoupling the revenue maximizationproblems in two parts: First, at each point in time thefirm selects an aggregate capacity consumption ratefrom all products, and second, it computes the vectorof demand rates to maximize instantaneous revenuessubject to the constraint that all products jointly con-sume capacity at the aforementioned rate. The latter isakin to the basic microeconomics problem of resourceallocation subject to a budget constraint, and givesrise to an appropriateaggregate revenue rate functionin each case. This common formulation recovers well-known structural results regarding the monotonicityproperties of the value function and the associatedcontrols (see Proposition 3 and Corollary 2), whichwere previously derived in the literature while study-ing each of these problems in isolation; see, ,Gallego and van Ryzin (1994, 1997), Lee and Hersh(1993), Lautenbacher and Stidham (1999), Zhao andZheng (2000), and the recent book by Talluri and vanRyzin (2004b).

6 It also facilitates the derivation of newresults, such as Corollary 1, that extend these proper-ties to the setting of multiproduct Pricing this idea of demand aggregation in con-sidering deterministic and continuous (fluid) approx-imations of the underlying problems suggests a setof simple Pricing and capacity control heuristics forthe underlying problems. These stem from the factthat this aggregated formulation leads to closed-form solutions to the fluid model revenue maxi-mization problems (see Proposition 4). This extendsthe analysis of Gallego and van Ryzin (1997), whichoffered only an implicit characterization of these poli-cies in the multiproduct setting; see also Bitran andCaldentey (2003) for a discussion of deterministicmultiproduct Pricing problems. Based on the solutionof the fluid formulation, we propose three heuristics:(i) a static Pricing heuristic, (ii) a static Pricing heuris-tic applied in conjunction with an appropriate capac-ity allocation policy, and (iii) a resolving heuristicthat reevaluates the fluid policy as a function of thecurrent state and time-to-go (which is derived byexpressing the fluid solution in feedback form).

7 Thefirst of these heuristics was proposed by Gallego andvan Ryzin (1997), while policies that combine staticprices with capacity controls as in (ii) have been sug-gested in other papers such as McGill and van Ryzin(1999), Feng and Xiao (2004), and Lin et al. (2003).Finally, the resolving heuristic (iii) is widely appliedin practice, but to the best of our knowledge hasnot been analyzed theoretically thus far. The onlyexception was the negative result of Cooper (2002),which illustrated through an example that resolvingmay in fact do worse than applying the static fluidpolicy. Propositions 5 7 in establishes that allthree heuristics achieve asymptotically optimal per-formance under fluid scaling, , in the spirit of Gal-lego and van Ryzin (1997) and Cooper (2002). Theseresults show that the phenomenon demonstrated inCooper s example does not persist in problems withlarge capacity and demand, where, in fact, resolv-ing achieves the asymptotically optimal , the numerical results of 5 illustrate thatthe Dynamic heuristics (ii) and (iii) tend to outperformthe static terms of qualitative insights, we find that thetranslation of the capacity consumption rate to a setof product-level controls that jointly maximize theinstantaneous revenue rate defines anefficient frontierfor the firm s optimal Pricing and capacity controlstrategies.

8 This captures in a tractable way the interac-tions between products due to cross-elasticity effectsand the joint capacity constraint. The idea of an effi-cient frontier has appeared in Talluri and van Ryzin(2004a) in the context of a capacity control problem fora model with customer choice among products, andin Feng and Xiao (2000, 2004) while studying pricingproblems with a predetermined set of price remainder of the paper is structured as fol-lows. This section concludes with some additionalbibliographic references. Section 2 describes the twoproblem formulations. Section 3 demonstrates thereduction of the Dynamic programming formulationsto that of a single-product Pricing problem, andderives some of its structural properties. Section 4studies the fluid formulations of these problems andanalyzes the asymptotic performance of the associ-ated heuristics mentioned earlier.

9 Section outlineshow the same approach can be extended to a networksetting. Section 5 provides some numerical illustrationof our results and offers some concluding idea of demand aggregation appeared in Tal-luri and van Ryzin (2004a) while analyzing a capac-ity control problem for a system where customerMaglaras and Meissner: Dynamic Pricing Strategies for Multiproduct Revenue Management Problems138 Manufacturing & Service Operations Management 8(2), pp. 136 148, 2006 INFORMS behavior is captured through a discrete choice model,while similar techniques have been exploited inthe past in the numerical solution of the dynamicprograms associated with revenue management prob-lems. Similar ideas also arise in problems of rev-enue management of multiproduct queueing sys-tems; see Maglaras (2006) for a recent papers by Elmaghraby and Keskinocak (2003),Bitran and Caldentey (2003), and McGill and vanRyzin (1999), and the book by Talluri and van Ryzin(2004b), provide comprehensive overviews of theareas of Dynamic Pricing and revenue modelling framework adopted in this paperclosely matches that of Gallego and van Ryzin (1994,1997).

10 Additional references on the capacity con-trol formulation are Brumelle and McGill (1993) andLautenbacher and Stidham (1999). Finally, a dynamicpricing heuristic that is related to the one studied inthis paper was derived by Reiman (2002) as the solu-tion to a second-order control problem that seeksto minimize an appropriate measure of the devia-tion from the inventory trajectory derived throughthe deterministic model. Indeed, the single-productpolicy proposed in Reiman (2002) is the same as theone we derive here. The multiproduct formulation inReiman (2002) differs from ours, and therefore theresulting Pricing policies also Single Resource, MultiproductModelThis section formulates the multiproduct dynamicpricing and capacity control problems studied byGallego and van Ryzin (1997) and Lee and Hersh(1993), respectively, among Dynamic Pricing a firmendowed withCunits of capacity of a single resourceused in producing or offering multiple products orservices, indexed byi=1 n.