MANUFACTURING & SERVICE
OPERATIONS MANAGEMENT informs ® Vol. 8, No. 2, Spring 2006, pp. 136–148 doi 10.1287/msom.1060.0105 issn 1523-4614 eissn 1526-5498 06 0802 0136 © 2006 INFORMS
Dynamic Pricing Strategies for Multiproduct Revenue Management Problems
Constantinos Maglaras Columbia Business School, Columbia University, 409 Uris Hall, 3022 Broadway, New York, New York 10027, [email protected] Joern Meissner Lancaster University Management School, Lancaster University, Room A48, Lancaster LA1 4XY, United Kingdom, [email protected]
onsider a firm that owns a fixed capacity of a resource that is consumed in the production or delivery of Cmultiple products. The firm strives to maximize its total expected revenues over a finite horizon, either by choosing a dynamic pricing strategy for each product or, if prices are fixed, by selecting a dynamic rule that controls the allocation of capacity to requests for the different products. This paper shows how these well- studied revenue management problems can be reduced to a common formulation in which the firm controls the 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 of the associated optimization problems. In the context of their associated deterministic (fluid) formulations, this reduction leads to a closed-form characterization of the optimal controls, and suggests several natural static and dynamic pricing heuristics. These are analyzed asymptotically and through an extensive numerical study. In the context of the former, we show that “resolving” the fluid heuristic achieves asymptotically optimal performance under fluid scaling. Key words: revenue management; dynamic pricing; capacity controls; fluid approximations; efficient frontier History: Received: July 21, 2003; accepted: November 11, 2005. This paper was with the authors 9 months for 3 revisions.
1. Introduction new requests for each of these products. In the sequel, Consider a firm that owns a fixed capacity of a cer- these two problems are referred to as the “dynamic tain resource that is consumed in the process of pro- pricing” and “capacity control” formulations, respec- ducing or offering multiple products or services, and tively. Revenue management problems of that sort which must be consumed over a finite time horizon. gained interest in the late 1970s in the context of The firm’s problem is to maximize its total expected the airline industry, and have since been successfully revenues by selecting the appropriate dynamic con- introduced in a variety of other areas such as hotels, trols. We consider two well-studied variants of this cruise lines, rental cars, retail, etc. problem. In the first, the firm is assumed to be a This paper illustrates how these two problems can monopolist or to operate in a market with imper- be reduced to a common formulation, thus connect- fect competition, and thus to have power to influ- ing prior results that have appeared in the literature ence the demand for each product by varying its under a unified framework, and explores some of price. In this setting, the firm’s problem is to choose the consequences of this formulation. Specifically, we a dynamic pricing strategy for each of its products show that the multiproduct dynamic pricing prob- to optimize expected revenues. In the second variant, lem introduced by Gallego and van Ryzin (1997) and prices are assumed to be fixed either by the competi- the capacity control problem of Lee and Hersh (1993) tion or through a higher-order optimization problem, can be recast within this common framework, and and the firm’s problem is now to choose a dynamic be treated as different instances of a single-product capacity allocation rule that controls when to accept pricing problem for appropriate concave revenue
136 Maglaras and Meissner: Dynamic Pricing Strategies for Multiproduct Revenue Management Problems Manufacturing & Service Operations Management 8(2), pp. 136–148, © 2006 INFORMS 137 functions (Propositions 1 and 2). Broadly speaking, (1999), Feng and Xiao (2004), and Lin et al. (2003). this is done by decoupling the revenue maximization Finally, the “resolving” heuristic (iii) is widely applied problems in two parts: First, at each point in time the in practice, but to the best of our knowledge has firm selects an aggregate capacity consumption rate not been analyzed theoretically thus far. The only from all products, and second, it computes the vector exception was the negative result of Cooper (2002), of demand rates to maximize instantaneous revenues which illustrated through an example that resolving subject to the constraint that all products jointly con- may in fact do worse than applying the static fluid sume capacity at the aforementioned rate. The latter is policy. Propositions 5–7 in §4.2 establishes that all akin to the basic microeconomics problem of resource three heuristics achieve asymptotically optimal per- allocation subject to a budget constraint, and gives formance under fluid scaling, i.e., in the spirit of Gal- rise to an appropriate aggregate revenue rate function lego and van Ryzin (1997) and Cooper (2002). These in each case. This common formulation recovers well- results show that the phenomenon demonstrated in known structural results regarding the monotonicity Cooper’s example does not persist in problems with properties of the value function and the associated large capacity and demand, where, in fact, resolv- controls (see Proposition 3 and Corollary 2), which ing achieves the asymptotically optimal performance. were previously derived in the literature while study- Moreover, the numerical results of §5 illustrate that ing each of these problems in isolation; see, e.g., the dynamic heuristics (ii) and (iii) tend to outperform Gallego and van Ryzin (1994, 1997), Lee and Hersh the static one. (1993), Lautenbacher and Stidham (1999), Zhao and In terms of qualitative insights, we find that the Zheng (2000), and the recent book by Talluri and van translation of the capacity consumption rate to a set Ryzin (2004b). It also facilitates the derivation of new of product-level controls that jointly maximize the results, such as Corollary 1, that extend these proper- instantaneous revenue rate defines an efficient frontier ties to the setting of multiproduct pricing policies. for the firm’s optimal pricing and capacity control Extending this idea of demand aggregation in con- strategies. This captures in a tractable way the interac- sidering deterministic and continuous (fluid) approx- tions between products due to cross-elasticity effects imations of the underlying problems suggests a set and the joint capacity constraint. The idea of an effi- of simple pricing and capacity control heuristics for cient frontier has appeared in Talluri and van Ryzin the underlying problems. These stem from the fact (2004a) in the context of a capacity control problem for that this aggregated formulation leads to closed- a model with customer choice among products, and form solutions to the fluid model revenue maxi- in Feng and Xiao (2000, 2004) while studying pricing mization problems (see Proposition 4). This extends problems with a predetermined set of price points. the analysis of Gallego and van Ryzin (1997), which The remainder of the paper is structured as fol- offered only an implicit characterization of these poli- lows. This section concludes with some additional cies in the multiproduct setting; see also Bitran and bibliographic references. Section 2 describes the two Caldentey (2003) for a discussion of deterministic problem formulations. Section 3 demonstrates the multiproduct pricing problems. Based on the solution reduction of the dynamic programming formulations of the fluid formulation, we propose three heuristics: to that of a single-product pricing problem, and (i) a static pricing heuristic, (ii) a static pricing heuris- derives some of its structural properties. Section 4 tic applied in conjunction with an appropriate capac- studies the fluid formulations of these problems and ity allocation policy, and (iii) a “resolving” heuristic analyzes the asymptotic performance of the associ- that reevaluates the fluid policy as a function of the ated heuristics mentioned earlier. Section 4.3 outlines current state and time-to-go (which is derived by how the same approach can be extended to a network expressing the fluid solution in feedback form). The setting. Section 5 provides some numerical illustration first of these heuristics was proposed by Gallego and of our results and offers some concluding remarks. van Ryzin (1997), while policies that combine static The idea of demand aggregation appeared in Tal- prices with capacity controls as in (ii) have been sug- luri and van Ryzin (2004a) while analyzing a capac- gested in other papers such as McGill and van Ryzin ity control problem for a system where customer Maglaras and Meissner: Dynamic Pricing Strategies for Multiproduct Revenue Management Problems 138 Manufacturing & Service Operations Management 8(2), pp. 136–148, © 2006 INFORMS behavior is captured through a discrete choice model, power to influence the demand for each product by while similar techniques have been exploited in varying its menu of prices. Let pt = p1t pnt the past in the numerical solution of the dynamic denote the vector of prices at time t. The demand programs associated with revenue management prob- process is assumed to be n-dimensional nonhomoge- lems. Similar ideas also arise in problems of rev- neous Poisson process with rate vector determined enue management of multiproduct queueing sys- through a demand function pt , where → , tems; see Maglaras (2006) for a recent example. ⊆ n is the set of feasible price vectors, and = n The papers by Elmaghraby and Keskinocak (2003), x ≥ 0x= p p ∈ ⊆ + is the set of achievable Bitran and Caldentey (2003), and McGill and van demand rate vectors. We assume that is a convex Ryzin (1999), and the book by Talluri and van Ryzin set. For ease of exposition, the demand function · (2004b), provide comprehensive overviews of the is assumed to be stationary; an extension to allow for areas of dynamic pricing and revenue management. nonstationarities could follow Gallego and van Ryzin The modelling framework adopted in this paper (1994, 1997) and Zhao and Zheng (2000). This class of closely matches that of Gallego and van Ryzin (1994, demand functions incorporates product complemen- 1997). Additional references on the capacity con- tarity and substitution effects. Following Gallego and trol formulation are Brumelle and McGill (1993) and van Ryzin (1994, 1997), we consider regular demand Lautenbacher and Stidham (1999). Finally, a dynamic functions that satisfy some additional conditions. In pricing heuristic that is related to the one studied in the sequel, x denotes the transpose of any matrix x, this paper was derived by Reiman (2002) as the solu- for any real number y, y+ = max0y , e is the vector tion to a “second-order” control problem that seeks of ones of appropriate dimension, and “a.s.” stands to minimize an appropriate measure of the devia- for almost surely. tion from the inventory trajectory derived through Definition 1. A demand function is said to be reg- the deterministic model. Indeed, the single-product ular if it is a continuously differentiable, bounded policy proposed in Reiman (2002) is the same as the function, and: (a) for each product i, ip is strictly one we derive here. The multiproduct formulation in decreasing in p , (b) lim p = 0 (i.e., consumers i pi→ i Reiman (2002) differs from ours, and therefore the have bounded wealth), and (c) the revenue rate resulting pricing policies also differ. n p p = i=1 pi ip is bounded for all p ∈ and has a finite maximizer p¯. 2. Single Resource, Multiproduct We assume that there exists a continuous inverse Model demand function p , p → , that maps an This section formulates the multiproduct dynamic achievable vector of demand rates into a corre- pricing and capacity control problems studied by sponding vector of prices p . This allows us to view Gallego and van Ryzin (1997) and Lee and Hersh the demand rate vector as the firm’s control, and infer (1993), respectively, among others. the appropriate prices using the inverse demand func- tion. The expected revenue rate can be expressed as 2.1.1. Dynamic Pricing Problem. Consider a firm a function of the vector of demand rates as R = endowed with C units of capacity of a single resource p , and is assumed to be continuous, bounded, used in producing or offering multiple products or and strictly concave. services, indexed by i = 1n. Each product i Example. Under a linear demand model, the de- request requires one unit of capacity. There is a finite mand for product i is given by horizon T over which the resources must be used, and capacity cannot be replenished up to that time. ip = i − biipi − bij pj or The salvage value of remaining capacity at time T j =i is assumed to be zero. (A constant per-unit sal- (in vector form) p = − Bp vage value would also result in formulations sim- ilar to those developed below.) The firm is either where i is the market potential for product i and bii, a monopolist or is assumed to operate in a mar- bij are the price and cross-price sensitivity parame- ket with imperfect competition, and, therefore, has ters. The inverse demand and revenue functions are Maglaras and Meissner: Dynamic Pricing Strategies for Multiproduct Revenue Management Problems Manufacturing & Service Operations Management 8(2), pp. 136–148, © 2006 INFORMS 139 p = B−1 − and R = B−1 − , respec- probability of accepting a product i request at time t. tively. Assumption 1 requires that bii > 0 for all i. It is customary to assume that the firm is “opening” To ensure that the inverse demand function is well or “closing” products, thus considering controls ui· defined and the revenue function is concave, we that are zero or one, but this need not be imposed as require that either bii > j =i bji or bii > j =i bij for a restriction. The dynamic capacity control problem is all i; both conditions guarantee that B is invertible and the following: that its eigenvalues have positive real parts (Horn and T T Johnson 1994, Thm. 6.1.10). max Ɛ p t u e t u ≤ C ut t=1T The problem that we address is roughly described t=1 t=1 as follows: Given an initial capacity C, a selling hori- a.s. 1 and u t ∈ 0 1 ∀ t (2) zon T , and a demand function that maps a vector of i prices to a vector of demand rates, the firm’s goal is to where u above denotes the vector with coordi- choose a nonanticipating dynamic pricing strategy for nates u . each product in order to maximize its total expected i i revenues. We adopt a discrete-time formulation, i.e., one 3. Analysis ofthe Pricing and where time has been discretized in small intervals Capacity Control Problems of length t, indexed by t = 1T, such that This section describes how to reduce (1) and (2) into dynamic optimization problems where the control is product i arrival in 0t = it + ot for all i, 2 the (one-dimensional) aggregate capacity consump- and product i and j arrivals in 0t = i j t + ot 2 , where ox implies that ox /x → 0asx → 0. tion rate. Subsequently, we derive some structural properties for these two problems through a unified With slight abuse of notation, we write i in place of analysis. it, and refer to i either as the demand or the buy- ing probability for product i. The random demand vector in period t, denoted by t , is Bernoulli 3.1. A Common Formulation in Terms of the Aggregate Capacity Consumption with probabilities t = pt , and it = 1 = ipt and it = 0 = 1− ipt for all i. Treating 3.1.1. Dynamic Pricing Problem. Let x denote the the demand rates i as the control variables (prices number of remaining units of capacity at the begin- are inferred via the inverse demand relationship), ning of period t, and Vxt be the expected revenue- the discrete-time formulation of the dynamic pricing to-go starting at time t with x units of capacity left. problem of Gallego and van Ryzin (1997) is Then, the Bellman equation associated with (1) is T n
max Ɛ p t t Vxt = max i pi + Vx − 1t+ 1 t t=1T ∈ t=1 i=1 T e t ≤C a.s. and t ∈ ∀t (1) + 1 − e Vx t + 1 (3) t=1 with the boundary conditions 2.1.2. Capacity Control Problem. The second problem that we consider is the one studied by VxT + 1 = 0 ∀ x and V0t = 0 ∀ t (4) Lee and Hersh (1993), where the price vector p and the demand rate vector = p are fixed, and the Letting !Vx t = Vxt + 1 − Vx − 1t+ 1 denote firm optimizes over capacity allocation decisions. For the marginal value of one unit of capacity as a func- this problem and without any loss of generality, we tion of the state xt , (3) can be rewritten as assume that products are labelled such that p1 ≥ n p ≥···≥p . The firm has discretion as to which prod- Vxt = max R − i!Vx t +Vxt + 1 2 n ∈ uct requests to accept at any given time. This is mod- i=1 = maxRr " − "!Vx t + Vxt + 1 (5) elled through the control uit that is equal to the "∈ Maglaras and Meissner: Dynamic Pricing Strategies for Multiproduct Revenue Management Problems 140 Manufacturing & Service Operations Management 8(2), pp. 136–148, © 2006 INFORMS
n where "= i=1 i is the aggregate rate of capacity con- Proposition 2. The capacity control problem (2) can n sumption, = " i=1 i = " ∈ is the set of be reduced to the dynamic program (9) and (4) expressed achievable capacity consumption rates, and in terms of the aggregate consumption rate ". In par- ticular, if "∗xt denotes the optimal solution of (9) n r and (4) and u∗xt denote the optimal policy for (2), then R " = max R i = " ∈ (6) ∗ a ∗ i=1 u xt = u " xt . is the maximum achievable revenue rate subject to the A similar result was derived by Talluri and van constraint that all products jointly consume capacity Ryzin (2004a) for a capacity control problem for a at a rate ". Note that (6) is a concave maximization model with customer choice. problem over a convex set, and its solution is read- ily computable, often in closed form (examples are 3.2. A Unified Analysis ofthe Pricing and given in §5). Moreover, Rr · is a concave function and Capacity Control Problems satisfies the conditions of Assumption 1. The optimal The preceding analysis illustrates that both problems vector of demand rates, denoted by r " , is unique can be reduced to “appropriate” single-product pric- and continuous in ". ing problems, highlighting their common structure and enabling a unified treatment. As a starting obser- Proposition 1. The dynamic pricing problem (1) can vation, for both (5) and (9) the optimal control "∗xt be reduced to the dynamic program (5) and (4) expressed is computed from in terms of the aggregate consumption rate. In particular, if "∗xt denotes the associated optimal control and ∗xt "∗xt = arg maxR" − "!Vx t and p∗xt denote the optimal demand rate and price vec- "∈ ∗ r ∗ tors associated with (1), then xt = " xt and where R· is a concave increasing revenue function. ∗ r ∗ p xt = p " x t . Using the properties of R· , one gets that "∗xt 3.1.2. The Capacity Control Problem. Similarly, is decreasing in !Vx t , which using a backward the Bellman equation associated with (2) is induction argument in t gives that !Vx t is decreas- ing in x and t. These standard results for single- n product dynamic pricing problems are summarized Vxt = max iui pi + Vx − 1t+ 1 u ∈ 0 1 i i=1 below; a proof can be found in Talluri and van Ryzin (2004b, Prop. 5.2, Ch. 4). + 1 − u Vx t + 1 (7) Proposition 3 (Talluri and van Ryzin 2004b, Prop. 5.2,Ch. 4). For both problems defined in (1) and (2), with the boundary condition (4), which, using the we have that marginal value of capacity !V becomes 1. "∗xt is decreasing in the marginal value of capac- n ity !Vx t , and
Vxt = max iuipi − u !Vx t u ∈ 0 1 2. !Vx t is decreasing in x and t. i i=1 + Vxt + 1 (8) Structural results for the pricing and capacity allo- r r a cation policies follow from the properties of R , = max R " −"!Vxt +Vxt +1 (9) a a n and R , u , respectively. For example, consider the 0≤"≤ i=1 i pricing problem for the case where the products are where " = u and nonsubstitutes, i.e., the demand for product i is only n a function of the price for that product pi. In that a R " = max ui ipiu = " ui ∈ 0 1 case, the Lagrangian associated with (6) is L x y = u i=1 n R + x" − i=1 i − y , with first-order condi- is the maximum revenue rate when the capacity is tions given by &R /& i = x + yi, for some x ≥ 0 a consumed at a rate equal to ", and u " is the corre- and yi ≤ 0 with yi = 0if i > 0. It is easy to show sponding control. that x is decreasing in " (i.e., the shadow price for Maglaras and Meissner: Dynamic Pricing Strategies for Multiproduct Revenue Management Problems Manufacturing & Service Operations Management 8(2), pp. 136–148, © 2006 INFORMS 141 the capacity consumption constraint decreases as the optimization, the efficient frontier provides a system- r rate " increases), and that i " is decreasing in x. atic framework for comparing different policies and highlights the structure of the respective optimal con- Corollary 1. Consider the problem specified in (3) trols. It may also lead to computational improve- and (4), and further assume that the products are non- ments if this subproblem can be solved efficiently, substitutes, i.e., p = p for all i. Then, ∗xt is i i i i preferably in closed form. This is possible in some nondecreasing in "∗xt (and nonincreasing in !Vx t ). common demand models such as the linear and the A similar result can be obtained when products are multinomial logit. Such an efficient frontier has also substitutable provided that the demand model satis- appeared and been discussed in more detail in Feng fies certain conditions analogous to those of the sen- and Xiao (2000, 2004) and Talluri and van Ryzin sitivity matrix B of the linear model in §2. (2004a). Finally, we note that the structure of the For the capacity control problem, it is easy to dynamic programs studied in this section has been recover some well-known structural properties of the observed in other papers, such as Lin et al. (2003) and optimal policy, see, e.g., Lee and Hersh (1993). Our their study of single-resource capacity control prob- derivation based on the capacity consumption rate lems where each arrival may request multiple units offers new intuition as to why they hold. Specifically, of capacity; and Vulcano et al. (2002) and their anal- Ra· is a knapsack solution for which ysis of optimal dynamic auctions. The latter involves an analysis of a discrete-time, batch-demand analog a R " = min ci + pi" and i to the dynamic program studied here. + (10) a " − i Our goal here is to analyze the “fluid-scale” behav- Analysis of the dynamic heuristic (c). The dynamic ior of the capacity process defined as Xkt = Xkt /k nature of this policy requires a more detailed study. under the three candidate policies. The asymptotic The cumulative demand for product i up to time t is optimality of the static policy (a) was established k equal to NiAi t , where by Gallego and van Ryzin (1997). Nevertheless, we t include our analysis, which is different than theirs k Ai t = k is ds where 0 and serves to introduce the ideas used in studying the r k dynamic policies (b) and (c). is = i min" Xc s /kT − s (20) Analysis of the static heuristic (a). The firm uses the for r · defined in (6), and Xkt denotes the remain- constant price vector p¯, which induces the demand i c rates kp ¯ = k ¯ . Under this policy, Akt = ¯ kt, and ing capacity at time t under policy (c), defined in (18). i i the capacity dynamics are Xkt = Ck − n a N · Now, note that Ak0 = 0, Akt is nondecreasing a i=1 i i k k k + and Ai t − Ai s ≤ t − s · k imax, where imax = Ai t . (The subscript is used to identify the policy.) As k → arg max i ∈ . This implies that the family of pro- k N Akt cesses 1/k A t is equicontinuous, and therefore i i → ¯ t a.s., uniformly in t ∈ 0T (19) relatively compact. It follows that it has a converging k i subsequence, say k , such that 1/k Akk t → A¯ t for from which it follows that as k → and for all j j i i all i. Along this subsequence we get that N Akj t /k t ∈ 0T i i j converges to A¯ t , and therefore that Xkj t itself con- n + i c k ¯ Xa t → C − ai it = C −¯"t a.s. verges to a limit x¯ct ; the last two results hold a.s., i=1 uniformly in t. Expression (22) will show that the (The · + was removed because from (13) "t¯ ≤ C for limit trajectories do not depend on the selection of all t ∈ 0T .) Let Rk denote the revenues extracted a the converging subsequence itself. In the sequel we k n ¯ under policy (a), and / = infs ≥ 0 i=1 aiNi iks denote the converging subsequence by k to simplify ≥ Ck be the random time where the aggregate r notation. Using the continuity of i · and Lemma 2.4 capacity requested reaches or exceeds the available of Dai and Williams (1995), we get that k k n ¯ ¯ k capacity C . Then, Ra = i=1 piNik i minT / − , 1 t Xks where is a random variable that corrects revenues Akt = r min " c ds k i i for the case where / capacity per request, and a linear demand relation- Table 1 Optimality Gaps Relative to Optimal Policy (DP) ship of the form p = − Bp with = 03 01 and C RevMax (%) Fluid (%) LPCC (%) DynPrice (%) DP T = 200 time periods. The price set is = p − Bp ≥ 0, the inverse demand is p = B−1 − , and 25 314 3.0 2.6 0.4 417.63 −1 30 222 4.4 3.3 1.0 440.53 the revenue function is R = B − . The poli- 35 126 5.1 1.91.3 451.63 cies that we consider are the following: 40 54 5.4 0.7 0.7 457.00 • “RevMax” corresponds to the monopoly price 45 18 1.8 0.2 0.2 459.50 vector p that maximizes the aggregate instantaneous 50 07 0.7 0.3 0.3 460.39 revenue rate disregarding the capacity constraints, Note. B = diag1 6 given by p = B−1 − ˆ , where ˆ = arg max B−1 · − ≥ 0 = 1/2 B−1 + B−1 −1. terms and was given by p = 03 − 1p and p = • “Fluid” implemented the price vector p¯ = p r " ¯ 1 1 2 01 − 6p . as defined in Proposition 4. 2 • “LPCC” is the joint (list) pricing and capacity The Fluid pricing problem becomes unconstrained control heuristic defined through (15). for C ≥ 40 units, in which cases the RevMax and • “DynPrice” is the dynamic implementation of the Fluid prices coincide. We observe the following. First, fluid policy defined through (16). the relative performance under all heuristics improves • “DP” implemented the solution of the dynamic as the capacity C increases; this is consistent with program outlined in §3. the results of Gallego and van Ryzin (1994, 1997) for Remark (Comments on Complexity). The expected Fluid and RevMax. Second, while the Fluid heuris- revenues under the first two static policies were com- tic that incorporates the capacity constraint outper- puted analytically using a binomial model that under forms RevMax when capacity is scarce, its regret over DP was simply VCT , while for the remaining the DP policy can still be substantial (2%–5%). Third, policies we averaged out revenues over 1,000 sim- when the capacity is scarce, C ≤ 25, the fluid prices ulated sample paths. The effort needed to compute effectively switch off Product 2 and operate the sys- VCT in problems with uniform capacity require- tem as a single-product one. As the capacity increases, ments (Tables 1 to 3) is that of solving a single- it is optimal to offer both products, and the effect product pricing problem, which grows in proportion of the capacity control of LPCC becomes more evi- to C × T .IfRr " and r " cannot be expressed in dent. Switching from an effectively single-product to closed form and are not precomputed and stored, a two-product solution causes the optimality gaps to the state space dimension of the dynamic program be nonmonotone. Fourth, the dynamic pricing (resolv- stays the same, but its complexity increases by a con- ing) heuristic is significantly better than all others stant factor that depends on the number of products. when capacity is scarce. Table 4 looks at examples with nonuniform capac- 5.1.2. The Cross-Price Elasticity Effects and Mul- ity requirements, where the demand aggregation no tiple Products. Table 2 summarizes results from 100 longer applies and the backward induction step of test cases for systems with fixed capacity C = 40, lin- the dynamic program needs to be changed, increasing ear demand model with = 03 01 , and randomly the overall complexity again by a constant factor. The complexity of computing LPCC and DynPrice, the performance of which is the emphasis of our study, is Table 2 Average and Standard Deviations of % Optimality Gaps for Two-Product Examples: C = 40, p = −Bp, = 03 01 , negligible. and Random B Matrices 5.1.1. The Effect of the Joint Capacity Constraint. RevMax Fluid LPCC DynPrice Table 1 provides an illustration of the behavior of 0.8–0.91.5 (0.3) 1.6 (0.3) 1.6 (0.3) 0.8 (0.2) these heuristics on a particular problem instance as 0.9–1.0 3.6 (0.6) 3.6 (0.5) 3.1 (1.6) 1.1 (0.2) we varied the available capacity. Tables 2 and 3 will 1.0–1.1 5.2 (1.1) 3.9(0.3) 3.0 (1.5) 1.0 (0.2) provide summary statistics for many randomly gener- 1.1–1.2 9.8 (1.3) 3.1 (0.4) 2.4 (0.3) 1.0 (0.2) ated test cases. The demand model had no cross-price 1.2–1.3 14.0 (1.5) 3.1 (0.4) 2.1 (0.2) 0.9(0.2) Maglaras and Meissner: Dynamic Pricing Strategies for Multiproduct Revenue Management Problems Manufacturing & Service Operations Management 8(2), pp. 136–148, © 2006 INFORMS 147 Table 3 Average and Standard Deviations of % Optimality Gaps for Table 4 Optimality Gaps Relative to Optimal Policy (DP) Three-Product Examples: C = 40, p = − Bp, = (i) B = 1 − 04 −066 and a = 12 03 005 005 , and Random B Matrices C RevMax (%) Fluid (%) LPCC (%) DynPrice (%) DP RevMax Fluid LPCC DynPrice 30 427 4.3 3.7 2.7 476.91 0.8–0.91.5 (0.3) 1.6 (0.2) 1.5 (0.3) 0.8 (0.1) 40 271 5.6 2.1 1.5 496.05 0.9–1.0 3.3 (0.7) 3.3 (0.7) 3.3 (1.6) 1.0 (0.2) 50 120 5.7 1.7 1.0 502.77 1.0–1.1 6.4 (1.4) 3.7 (0.4) 3.6 (2.0) 1.1 (0.2) 60 293.2 1.1 0.3 505.81 1.1–1.2 9.4 (1.3) 3.1 (0.3) 3.0 (2.0) 0.9 (0.2) 70 08 0.7 0.4 0.3 506.73 1.2–1.3 14.0 (1.3) 3.0 (0.3) 2.1 (1.5) 0.8 (0.2) 80 03 0.2 0.2 0.1 506.86 generated B matrices; the elements of the main diago- (ii) B = 1 − 04 −066 and a = 21 nal were drawn from a uniform distribution on 0 1 , C RevMax (%) Fluid (%) LPCC (%) DynPrice (%) DP while the cross terms were drawn from a uniform dis- 30 395 4.4 4.0 3.9340.95 tribution on −1 0 . We then tested that the dominant 40 322 3.3 3.0 1.7 406.89 diagonal condition given in §2 was satisfied, which 50 234 3.0 2.90.6 451.85 also assured that an inverse exists. Results have been 60 142 3.7 3.4 1.2 479.64 grouped according to their nominal load factors " = 70 60 4.4 3.6 0.8 495.78 ˆ ˆ 80 17 2.0 1.7 0.7 503.60 i ai i /C, where is the RevMax demand vector that would maximize instantaneous revenue in the absence of the capacity constraint.1 The table reports that the fluid model heuristics perform quite well in the average and standard deviation of the % optimal- cases where the capacity requirements are small com- ity gap for each candidate policy. The main obser- pared to the capacity itself. (Eventually, as C and vation is that DynPrice significantly outperformed grow large, these policies become asymptotically opti- all other heuristics not only in terms of the average mal in the sense of Propositions 5 to 7.) gap, but also in its standard deviation. In turn, LPCC improved over Fluid in cases where the capacity con- The last three tables illustrate that static pricing is straint was binding and the ability to incorporate close to optimal in problem instances that are either the extra element of capacity control decisions was very under- or overloaded, and that the effect of beneficial. Table 3 reports results for three-product dynamic pricing, either through the optimal policy examples, and illustrates the consistently good perfor- (DP) or the DynPrice heuristic, is most pronounced mance of the LPCC and DynPrice policies. The resolv- when " is close to 1. This is intuitive: When " 1, ing structure of the latter is important in problems the system’s capacity far exceeds the nominal require- with multiple products and substitution and/or com- ment under the prices that maximize revenues per plementarity effects. unit time (implemented in RevMax), and thus one would expect that the static pricing heuristics RevMax 5.1.3. Nonuniform Capacity Requirements. Table and Fluid to be close to optimal. In contrast, if " 1, 4 summarizes results for two models with two prod- the system’s capacity is too low, and the static prices ucts that have different capacity requirements. This implemented through the Fluid heuristic are close change complicates the associated dynamic program- to optimal. Finally, we did not observe any problem ming formulation (see comments in the beginning of this section), but does not affect the fluid analy- instances where the performance under the DynPrice sis of §4.1, and the heuristics extracted therein are heuristic was worse (in a statistically significant man- still valid. In the notation of §4, product i consumes ner) than that under the LPCC or Fluid heuristics. The narrowest gap between LPCC and DynPrice was ai units of capacity and a1 = a2. 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