In this paper, we propose a new dynamic programming decomposition method for the network revenue management problem with customer choice behavior. The fundamental idea behind our dynamic programming decomposition method is to allocate the revenue associated with an itinerary among the different flight legs and to solve a single-leg revenue management problem for each flight leg in the airline network. The novel aspect of our approach is that it chooses the revenue allocations by solving an auxiliary optimization problem that takes the probabilistic nature of the customer choices into consideration. We compare our approach with two standard benchmark methods. The first benchmark method uses a deterministic linear programming formulation. The second benchmark method is a dynamic programming decomposition idea that is similar to our approach, but it chooses the revenue allocations in an ad hoc manner. We establish that our approach provides an upper bound on the optimal total expected revenue, and this upper bound is tighter than the ones obtained by the two benchmark methods. Computational experiments indicate that our approach provides significant improvements over the performances of the benchmark methods.

We propose a new method for making the inventory replenishment decisions in distribution systems. In particular, we consider distribution systems consisting of multiple retailers that face random demand and a warehouse that supplies the retailers. The method that we propose is based on formulating the distribution problem as a dynamic program and relaxing the constraints that ensure the nonnegativity of the shipments to the retailers by associating Lagrange multipliers with them. We show that our method provides lower bounds on the value functions and a good set of values for the Lagrange multipliers can be obtained by maximizing a concave function in a relatively straightforward manner. Computational experiments indicate that our method can provide significant improvements over the traditional approaches for making the inventory replenishment decisions, in terms of both the tightness of the lower bounds on the value functions and the performance of the policies.

We present a new deterministic linear program for the network revenue management problem with customer choice behavior. The novel aspect of our linear program is that it naturally generates bid prices that depend on how much time is left until the time of departure. Similar to the earlier linear program used by van Ryzin and Liu (2004), the optimal objective value of our linear program provides an upper bound on the optimal total expected revenue over the planning horizon. In addition, the percent gap between the optimal objective value of our linear program and the optimal total expected revenue diminishes in an asymptotic regime where the leg capacities and the number of time periods in the planning horizon increase linearly with the same rate. Computational experiments indicate that when compared with the linear program that appears in the existing literature, our linear program can provide tighter upper bounds and the control policies that are based on our linear program can obtain higher total expected revenues.

In this paper, we consider numerous inventory control problems for which the base-stock policies are known to be optimal, and we propose stochastic approximation methods to compute the optimal base-stock levels. The existing stochastic approximation methods in the literature guarantee that their iterates converge, but not necessarily to the optimal base-stock levels. In contrast, we prove that the iterates of our methods converge to the optimal base-stock levels. Moreover, our methods continue to enjoy the well-known advantages of the existing stochastic approximation methods. In particular, they only require the ability to obtain samples of the demand random variables, rather than to compute expectations explicitly, and they are applicable even when the demand information is censored by the amount of available inventory.

The network revenue management (RM) problem arises in airline, hotel, media, and other industries where the sale products use multiple resources. It can be formulated as a stochastic dynamic program, but the dynamic program is computationally intractable because of an exponentially large state space, and a number of heuristics have been proposed to approximate its value function. In this paper we show that the piecewise-linear approximation to the network RM dynamic program is tractable; specifically we show that the separation problem of the approximation can be solved as a relatively compact linear program. Moreover, the resulting compact formulation of the approximate dynamic program turns out to be exactly equivalent to the Lagrangian relaxation of the dynamic program, an earlier heuristic method proposed for the same problem. We perform a numerical comparison of solving the problem by generating separating cuts or as our compact linear program. We discuss extensions to versions of the network RM problem with overbooking as well as the difficulties of extending it to the choice model of network revenue RM.