Re-Solving Stochastic Programming Models for Airline Revenue Management

We study some mathematical programming formulations for the origin-destination model in airline revenue management. In particular, we focus on the traditional probabilistic model proposed in the literature. The approach we study consists of solving a sequence of two-stage stochastic programs with simple recourse, which can be viewed as an approximation to a multi-stage stochastic programming formulation to the seat allocation problem. Our theoretical results show that the proposed approximation is robust, in the sense that solving more successive two-stage programs can never worsen the expected revenue obtained with the corresponding allocation policy. Although intuitive, such a property is known not to hold for the traditional deterministic linear programming model found in the literature. We also show that this property does not hold for some bid-price policies. In addition, we propose a heuristic method to choose the re-solving points, rather than re-solving at equally spaced times as customary. Numerical results are presented to illustrate the effectiveness of the proposed approach

Solving the vehicle routing problem with stochastic demands using the cross-entropy method.

Abstract An alternate formulation of the classical vehicle routing problem with stochastic demands (VRPSD) is considered. We propose a new heuristic method to solve the problem. The algorithm is a modified version of the so-called Cross-Entropy method, which has been proposed in the literature as a heuristic for deterministic combinatorial optimization problems based upon concepts of rare-event simulation. In our version of the method, the objective function is computed using Monte-Carlo simulations at each point in the domain and the modified CrossEntropy heuristic is applied. A framework is also developed for obtaining exact solutions and tight lower bounds for the problem under various conditions, which include specific families of demand distributions. This is used to assess the heuristic's performance. Finally, numerical results are presented for various problem instances to illustrate the ideas

On Rate of Convergence of Optimal Solutions of Monte Carlo Approximations of Stochastic Programs

In this paper we discuss Monte Carlo simulation based approximations of a stochastic programming problem. We show that if the corresponding random functions are convex piecewise smooth and the distribution is discrete, then (under mild additional assumptions) an opitmal solution of the approximating problem provides an exact optimal solution of the true problem with probability one for sufficiently large sample size. Moreover, by using theory of Large Deviations, we show that the probability of such an event approaches one exponentially fast with increase of the sample size. In particular, this happens in the case of two stage stochastic programming with recourse if the corresponding distributions are discrete. The obtained results suggest that, in such cases, Monte Carlo simulation based methods could be very efficient. We present some numerical examples to illustrate the involved ideas

Variable-sample methods for stochastic optimization

In this article we discuss the application of a certain class of Monte Carlo methods to stochastic optimization problems. Particularly, we study variable-sample techniques, in which the objective function is replaced, at each iteration, by a sample average approximation. We first provide general results on the schedule of sample sizes, under which variable-sample methods yield consistent estimators as well as bounds on the estimation error. Because the convergence analysis is performed pathwisely, we are able to obtain our results in a flexible setting, which requires mild assumptions on the distributions and which includes the possibility of using different sampling distributions along the algorithm. We illustrate these ideas by studying a modification of the well-known pure random search method, adapting it to the variable-sample scheme, and show conditions for convergence of the algorithm. Implementation issues are discussed and numerical results are presented to illustrate the ideas. Categories and Subject Descriptors: G.1.6 [Numerical Analysis]: Optimization—global optimization

A Study on the Cross-Entropy Method for Rare-Event Probability Estimation

We discuss the problem of estimating probabilities of rare events in static simulation models using the recently proposed cross-entropy method, which is a type of importance-sampling technique in which the new distributions are successively calculated by minimizing the crossentropy with respect to the ideal (but unattainable) zero-variance distribution. In our approach, by working on a functional space we are able to provide an efficient procedure without assuming any specific family of distributions. We then describe an implementable algorithm that incorporates the ideas described in the paper. Some convergence properties of the proposed method are established, and numerical experiments are presented to illustrate the efficacy of the algorithm

Variable-Sample Methods and Simulated Annealing for Discrete Stochastic Optimization

In this paper we discuss the application of a certain class of Monte Carlo methods to stochastic optimization problems. Particularly, we study variable-sample techniques, in which the objective function is replaced, at each iteration, by a sample average approximation. We first provide general results on the schedule of sample sizes, under which variable-sample methods yield consistent estimators as well as bounds on the estimation error. Because the convergence analysis is performed sample-path wise, we are able to obtain our results in a flexible setting, which includes the possibility of using different sampling distributions along the algorithm, without making strong assumptions on the underlying distributions. In particular, we allow the distributions to depend on the decision variables x. We illustrate these ideas by studying a modification of the wellknown simulated annealing method, adapting it to the variable-sample scheme, and show conditions for convergence of the algorithm

Simulation-based methods for stochastic optimization

Ph.D.Alexander Shapir