A firm has inventories of a set of components that are used to produce a set of products. There is a finite horizon over which the firm can sell its products. Demand for each product is a stochastic point process with an intensity that is a function of the vector of prices for the products and the time at which these prices are offered. The problem is to price the finished products so as to maximize total expected revenue over the finite sales horizon. An upper bound on the optimal expected

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... nue is established by analyzing a deterministic version of the problem. The solution to the deterministic problem suggests two heuristics for the stochastic problem that are shown to be asymptotically optimal as the expected sales volume tends to infinity. Several applications of the model to network yield management are given. Numerical examples illustrate both the range of problems that can be modeled under this framework and the effectiveness of the proposed heuristics. The results provide several fundamental insights into the performance of yield management systems. GALLEGO AND VAN RYZIN / 25 travel restrictions, etc., then closing a fare class is mathematically equivalent to setting the price for the corresponding product at a sufficiently high price, so that the expected demand rate is zero, and the fare class is effectively closed. In practice, of course, one would simply tell customers that the fare is no longer available. By modeling the closing of fares as we do, with no additional demands occurring after the last sale, we intentionally blur the distinction between demand and sales, and view pricing and allocation as one decision. In Section 7.2.1, we discuss situations in which limitations on pricing flexibility require a separate allocation policy. Second, if one offers multiple fare classes for a given product which is not well differentiated, closing a low fare is mathematically equivalent to setting it to the next highest fare. Notice that this may cause an increase in demand at the next highest fare because some customers who are willing to buy at the low fare are also willing to buy at the next highest fare when the former is not available. This phenomenon is known in the airline industry as demand recapture. In both instances, closing fare classes is equivalent to changing the "menu" of prices offered to customers at any point in time. In these two ways, one can map a wide range of joint pricing/allocation schemes into equivalent purepricing schemes. The advantage of the pricing framework is that it allows for a unified analysis of a rich class of yield management problems, and it directly reveals the relationship between pricing and allocation decisions. Multidimensional problems of the type suggested by our formulation arise in a wide variety of applications. For example, in the case of airlines, a "product" is an itinerary (path) from an origin (0) to a destination (D) in an airline network and a "resource" is a seat on a particular flight leg (edge) in the network. A customer traveling from 0 to D requires one unit of resource on each leg in a particular itinerary from 0 to D. In some cases, one might have two products that use precisely the same set of resources. Again, this occurs in airline applications when well differentiated coach and super-saver products exist for each itinerary. In this case a product is an O-D itinerary at a certain fare level/restriction combination. Another example of such multidimensional problems is found in the hotel industry. Yield management problems for hotels are often viewed as quite different from airline problems since customers may occupy a room for several nights. Because of this overlapping effect, each day cannot be viewed as an independent instance. To model this problem under the above framework, one defines a product for each possible combination of days that a customer might request over a specific period of time, e.g., one month. Resources correspond to room capacities on each day of the planning period (perhaps net of any group reservations), and products correspond to particular subsets of days. Fior example, one product could be a Monday-to-Thursday stay during Week 1. If sold, this product would then consume one unit each of the resources correspond- / GALLEGO AND VAN RYZIN Proof. Proceed by applying Jensen's inequality to the third term of JU(x, t) in (20) and by viewing the integrand inside the expectation as purely a function of A and maximizing pointwise. The result then follows as before.