A MULTIPRODUCTDYNAMIC PRICING PROBLEM AND ITS APPLICATIONSTO NETWORKYIELD MANAGEMENT GUILLERMOGALLEGO and GARRETTVAN RYZIN Columbia University,New York (ReceivedMay 1993;revisions received October 1993, November 1993; accepted September 1994) A firmhas inventoriesof a set of componentsthat are used to producea set of products.There is a finitehorizon over which the firm can sell its products.Demand for each productis a stochasticpoint processwith an intensitythat is a functionof the vectorof prices for the productsand the time at whichthese pricesare offered.The problemis to price the finishedproducts so as to maximizetotal expected revenue over the finite sales horizon.An upper bound on the optimal expected revenue is establishedby analyzinga deterministicversion of the problem.The solutionto the deterministicproblem suggests two heuristicsfor the stochasticproblem that are shown to be asymptoticallyoptimal as the expected sales volume tends to infinity.Several applicationsof the model to networkyield managementare given. Numericalexamples illustrate both the range of problemsthat can be modeled under this frameworkand the effectivenessof the proposedheuristics. The resultsprovide several fundamental insights into the performanceof yield managementsystems. yield management is the practice of using booking decision in a multiple-market, multiple-resource setting. policies together with information systems data to Specifically, we define the problem in terms of products increase revenues by intelligently matching capacity with and resources.To provide a unit of product j, j = 1, . .. , n demand. It is now widely practiced in capacity-constrained requires aij units of resource i, i = 1,..., m. We define service industries such as the airlines, hotels, car rentals, the bill of materialsmatrix A = [aij]. We assume A is an and cruise-lines. Historically,yield management started as integer valued matrix.Demand for each product j is mod- an operations function, focusing only on capacity alloca- eled as a stochastic point process with intensity AJ.(All tion given exogenousdemand estimates. However, there is vectors are column vectors unless otherwise stated.) The a growing consensus among researchers and practitioners vector of demand intensities A = (A1,. ., An)at time s, is alike that the pricing decisions that induce demand cannot a function of s and the vector of current prices p = be separated from traditional,capacity-oriented yield man- (pt, ... , pn). This demand function need only satisfy some agement decisions; these two decision are inextricably general regularityconditions. Demand for a given product linked. Pricing decisions influence the demand statistics j is allowed to depend on time, the price of product j, and that form the basic inputs to any yield management sys- the price of products other than j; however, we do not tem, and pricing decisions in turn must ultimatelybe based allow the currentdemand to be a function of past or future on knowledge of the underlying capacity constraints and prices. Thus, we assume that customers respond only to yield management policies. Therefore, to realize the full current prices and do not act strategically by adjusting benefits of yield management, joint pricing/allocation their buying behavior in response to the firm's pricing pol- schemes are necessary. icy. This is certainly a limitation of the model. To consider While what must be done, namely coordinating pricing strategic customer behavior would require a game- and allocation, is clear, exactly how it should be accom- theoretic formulation, which, though quite interesting, is plished remains somewhat a mystery.The problem is com- beyond the scope of our present analysis. plicated by the fact that in many applications,most notably Given initial quantities of resources at time zero and a the airlines, many markets (origin-destinationpairs) com- time horizon t, the problem is to control prices, perhaps pete for the same resources (leg capacities); therefore, subject to price constraints,in order to maximize expected the pricing/allocation problem must be solved jointly for revenue over the interval [0, t]. a large number of markets and resources. Uncertainty in It is perhaps not immediatelyclear that this formulation the demand process and the dynamic nature of the allo- is a reasonable model of a yield management problem. In cation decisions further compound the difficulty of the problem. particular, there appear to be no "allocation" decisions. However, one can view allocation, at least conceptually,as a pricing decision. 1. MODEL DESCRIPTION AND APPLICATIONS This is true for two reasons. First, if each product has We propose a dynamic, stochastic model of yield manage- only one fare class associated with it or each fare class is ment problems that addresses the joint pricing/allocation well differentiated in terms of payment requirements, Subject classifications: Transportation: yield management, seat allocation, pricing. Dynamic programming: intensity control. Area of review: STOCHASTICPROCESSES & THEIR APPLICATIONS. Operations Research 0030-364X/97/4501-0024 $05.00 Vol. 45, No. 1, January-February 1997 24 ? 1997 INFORMS GALLEGO AND VAN RYZIN / 25 travel restrictions, etc., then closing a fare class is mathe- ing to the days Monday,Tuesday, Wednesday, and Thursday matically equivalent to setting the price for the corre- of Week 1. Our multiproduct,multiresource formulation nat- sponding product at a sufficientlyhigh price, so that the urally captures the inter-day dependencies found in hotel expected demand rate is zero, and the fare class is effec- booking. tively closed. In practice, of course, one would simply tell Our model also applies to situations quite distinct from customers that the fare is no longer available.By modeling those of traditionalyield management. One example is the the closing of fares as we do, with no additional demands style or fashion goods industry. In the fashion industry, occurring after the last sale, we intentionally blur the dis- contracts for garment production typicallyare fixed well in tinction between demand and sales, and view pricing and advance of a given selling season. During a sales season, allocation as one decision. In Section 7.2.1, we discuss retailers face the problem of adjusting prices, usually situations in which limitations on pricing flexibilityrequire through periodic sales and/or mark-downs,so as to maxi- a separate allocation policy. mize the total revenue from their initial stock. These prob- Second, if one offers multiple fare classes for a given lems are often multi-dimensional in nature for two product which is not well differentiated,closing a low fare reasons. First, demand for products may be dependent due is mathematicallyequivalent to setting it to the next high- to the fact that one product may substitute for or comple- est fare. Notice that this may cause an increase in demand ment another. Second, both products and resources may at the next highest fare because some customers who are correspond to specific garment types at particularlocations willing to buy at the low fare are also willing to buy at the (either retail outlets or warehouses). Thus, if transship- next highest fare when the former is not available. This ments are allowed, the same "product"(garment j at loca- phenomenon is known in the airline industry as demand tion x) may be provided using one of several "resources" recapture. (garment j at another location y). In both instances, closing fare classes is equivalent to Other possible applications of the model include pro- changing the "menu"of prices offered to customers at any duction/inventoryproblems where resources are compo- point in time. In these two ways, one can map a wide range nents that can be assembled into a certain number of of joint pricing/allocation schemes into equivalent pure- available products in a product line. The problem then is pricing schemes. The advantage of the pricing framework to price the product line to maximize the revenue from a is that it allows for a unified analysisof a rich class of yield given initial stock of components. For example, assembling management problems, and it directly reveals the relation- a line of personal computers which share common compo- between and ship pricing allocation decisions. nents but are subject to rapid obsolescence is one possible Multidimensionalproblems of the type suggested by our application in this category. formulation arise in a wide variety of applications. For In the interest of brevity and to maintain focus, we shall example, in the case of airlines, a "product"is an itinerary concentrate on airline yield management applications of (path) from an origin (0) to a destination (D) in an airline our model. However, both our model and analysis are in- network and a "resource"is a seat on a particularflight leg deed quite general and can be used in a wide variety of (edge) in the network. A customer traveling from 0 to D other applications. requires one unit of resource on each leg in a particular itineraryfrom 0 to D. In some cases, one might have two products that use precisely the same set of resources. 2. LITERATUREREVIEW Again, this occurs in airline applications when well differ- entiated coach and super-saverproducts exist for each itin- The study of yield management problems in the airlines erary. In this case a product is an O-D itinerary at a dates back to the work of Rothstein (1971) for an over- certain fare level/restrictioncombination. booking