Computing Time-Dependent Bid Prices in Network Revenue Management Problems

We propose a new method to compute bid prices in network revenue management problems. The novel aspect of our method is that it naturally provides dynamic bid prices that depend on how much time is left until departure. We show that our method provides an upper bound on the optimal total expected revenue and this upper bound is tighter than the one provided by the widely known deterministic linear programming approach. Furthermore, it is possible to use the bid prices computed by our method as a starting point in a dynamic programming decomposition-like idea to decompose the network revenue management problem by the flight legs and to obtain dynamic and capacity-dependent bid prices. Our computational experiments indicate that the proposed method improves on many standard benchmarks. The idea of bid prices forms a powerful tool for building good and practical policies for network revenue management problems. This idea associates a bid price with each flight leg that captures the opportunity cost of a unit of capacity. An itinerary request is accepted only when there is enough capacity and the revenue from the itinerary request exceeds the sum of the bid prices associated with the flight legs that are in the requested itinerary; see Williamson (1992) and Talluri and van Ryzin (1998). Although it is known that the optimal policies are not necessarily characterized by bid prices, the intuitive appeal and ease of implementation of the bid price policies make them a popular choice in practice. One of the traditional approaches for computing bid prices is based on solving a deterministic linear program. This linear program can be viewed as a deterministic approximation to the network revenue management problem that is formulated under the assumption that the numbers of itinerary requests are known in advance and they take on their expected values. In the deterministic linear program, there exists one constraint for each flight leg and the right sides of these constraints are the remaining leg capacities. Therefore, the optimal values of the dual variables associated with these capacity constraints are used as bid prices. Nevertheless, since the deterministic linear program assumes that the numbers of itinerary requests are known in advance, it does not accurately capture the temporal dynamics of the arrivals of the itinerary requests. In this paper, we propose a new method to compute dynamic bid prices that depend on how much time is left until the time of departure. Our goal is to compute bid prices that capture the temporal dynamics of the arrivals of the itinerary requests somewhat more accurately than the deterministic linear program. The linear program described above can be viewed as a deterministic approximation, whereas our method directly works with the dynamic programming formulation of the network revenue management problem. The main idea behind our method is to relax the capacity constraints in the dynamic programming formulation by associating Lagrange multipliers with them. In this case, the optimality equation decomposes by the time periods and we obtain a simple expression for the value function. We show that a good set of values for the Lagrange multipliers can be obtained by solving a linear program. Since our Lagrange multipliers depend on how much time is left until departure, the bid prices that we obtain also depend on how much time is left until departure. Our method shares many of the appealing features of the traditional linear programming approach for computing bid prices. To begin with, it is known that the optimal objective value of the deterministic linear program provides an upper bound on the optimal total expected revenue and we show that our method also provides such an upper bound. Furthermore, a popular method to compute dynamic bid prices is based on decomposing the network revenue management problem into a sequence of single leg revenue management problems. This idea is known as dynamic programming decomposition and the bid prices computed by the deterministic linear program serve as a starting point for this method; see Section 3.4.4 in Talluri and van Ryzin (2004). As a matter of fact, it is possible to show that dynamic programming decomposition computes dynamic and capacity-dependent bid prices that depend not only on how much time is left until departure, but also on how many units of capacity are left on the flight legs. We show that the bid prices computed by our method can also be used in a dynamic programming decomposition-like idea to compute dynamic and capacity-dependent bid prices. Finally,