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Article Excerpt
To optimize revenue, service firms must integrate within their pricing policies the rational of customers to their price schedules. In the airline or telecommunication industry, this process is all the more complex due to interactions resulting from the structure of the supply network. In this paper, we consider a streamlined version of this situation where a firm's decision variables involve both prices and investments. We model this situation as a joint design and pricing problem that we formulate as a mixed-integer bilevel program, and whose properties are investigated. In particular, we take advantage of a feature of the model that allows the development of an algorithmic framework based on Lagrangean relaxation. This approach is entirely novel, and numerical results show that it is capable of solving problems of significant sizes.
Subject classifications: economics: pricing; games; integer programming: heuristics; transportation; programming: nonlinear, nondifferentiable.
Area of review: Transportation.
History: Received June 2004; revisions received December 2005, August 2006; accepted September 2006.
1. Introduction
This paper is devoted to a model that captures the inter-action between system design, price setting, and consumer choice over a transportation network, without assuming the a priori knowledge of demand functions. The problem involves two decision makers acting noncooperatively and in a sequential way. The upper level (leader) strives to maximize its revenue raised from tariffs imposed on a set of goods or services in its control, whereas the lower level (follower) optimizes its own objective, taking into account the tariff schedule set by the leader. The leader explicitly incorporates the reaction of the follower in his optimization process. In the field of economics, this fits the principal/agent paradigm (Van Ackere 1993) where the principal, fully aware of the agent's rational behaviour, induces cooperation from the agent through an incentive scheme. In the field of mathematical programming, this problem belongs to the class of bilevel optimization problems with bilinear objectives at both levels of decision.
In the current context of deregulation, pricing decisions have become crucial for airline, trucking, telecommunication, and service industries where intense price competition and network modifications have occurred. Clearly, a profit-maximizing firm must consider the trade-off between the cost of service and the revenue generated when designing its system and prices.
In the passenger or freight airline industry, a carrier (the leader) selects routing patterns, flight schedules, and fares. For instance, Budenbender et al. (2000) describe a system where freight providers such as express shipment companies operate or rent an aircraft fleet that must provide a high level of service. For consolidation purposes, the freight is first shipped to an airport. Next, it is flown nonstop to another airport, finally to be loaded on trucks and shipped to its final destination. The problem then consists of deter-mining the terminal to operate, the take-off time, how to transport the freight to an airport, and the rate to charge. In passenger transportation, the introduction of new flights (direct or through a hub-and-spoke network) must take into account the supply over the entire network of flights, both from the leader airline and its competitors. The decisions are then taken with respect to the incurred costs, the quality of service, the possible influence on demand to other destinations and, most important, the revenues generated by the new services (Lederer 1993, Lederer and Nambimadon 1998).
In the surface freight transportation industry, important structural changes occur as shippers optimize the end-to-end supply through the implementation of Web-based portals. In that context, the costs incurred by a carrier is made up of two components: a fixed cost (including trade compliance, trade settlement with country-specific international trading portals, multimodal aspects, operating resources costs, global handling costs, etc) and a unit transportation cost (Kerr 2001). Upon reception, a service carrier (the leader) has to decide whether or not to accept a request and, if accepted, to set a price. In reaction to those prices, the shippers (the follower) want their goods to be transshipped at minimal cost, hence the bilevel structure of the problem.
In the telecommunication area, a service provider (the leader) has to make network deployment decisions and to set prices for bandwidth usage. The response of users (the follower) to prices induces traffic on the network. In the current deregulated markets, pricing is a fundamental issue for communications carriers. Indeed, as new systems of ever-larger capacities are introduced, the marginal cost of data transmission is rapidly decreasing. Exploiting those cost savings and handling increased demand involves the optimization of technology acquisition and pricing processes (Lanning et al 2000, Basar and Srikant 2002). A recent paper by Bienstock et al. (2006) addresses this issue within a dynamic framework, assuming a demand model involving constant elasticities and null cross elasticities. Several references to network design problems can be found therein, and also in the classic book by Ahuja et al.(1993).
Design and pricing are also challenging issues for business information service providers (Bashyam 2000). Information agencies such as Reuters and Bloomberg (foreign currency markets) and Aspect Development (component information services) are essentially intermediaries between firms that generate, and firms that use, content. Because information service providers (the leader) incur large fixed costs (data entry and updates, software development, database management systems, connections to commercial networks), their problem consists of specifying the size of the database they provide to subscribers (followers) as well as the price they will charge for subscriptions. At the lower level, the subscribers adapt their usage volume according to the level of service and tariffs of the service providers, or may select the self-service option whereby they collect and collate information directly from the sources.
Until now, design and pricing issues have mostly been treated separately. However, they are intrinsically linked and have to be addressed jointly. To our knowledge, the only papers addressing the joint design and pricing problem are those of Lederer (1993), Basar and Srikant (2002), and Bashyam (2000). Lederer (1993) proposes a Nash equilibrium model of air transport competition where firms select routes and prices. Competition is studied under two different assumptions about consumer choice: Either consumers can spread their choice route using links belonging to different firms ("bundling" in the sense of Lederer), or they cannot. If bundling is forbidden, the author proves the existence of unique equilibrium prices. Otherwise, a price equilibrium may fail to be unique, or even to exist. At first glance, our work might seem to fit the frame-work analyzed by Lederer. However it differs in two main respects: bundling is an essential part of our model, and we look for a Stackelberg (leader-follower) equilibrium rather than a Nash equilibrium. Consequently, the focus of this paper is on algorithmic development rather than on economic considerations.
Basar and Srikant (2002) study the economics of providing large capacity from a telecommunication provider's point of view. Design choices are not modelled using binary decisions, but through continuous-capacity variables. Each user is charged a fixed price per unit of bandwidth used, and this price is independent from congestion. The transmission rate of each user is assumed to be a function or network congestion and price per unit of bandwidth. The aim of the service provider is to maximize its revenue. The authors show that, as the number of users increases, the optimal price per unit of bandwidth charged by the service provider may increase or decrease depending upon the bandwidth of the link. However, for all values of the link capacity, the overall performance of each user improves and the service provider's revenue per unit of bandwidth increases, thus providing an incentive for the service provider to increase the available bandwidth in proportion to traffic. Although this...
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