Shelby L. Brumelle

Airline seat allocation with multiple nested fare classes

This paper addresses the problem of determining optimal booking policies for multiple fare classes that share the same seating pool on one leg of an airline flight when seats are booked in a nested fashion and when lower fare classes book before higher ones. We show that a fixed-limit booking policy that maximizes expected revenue can be characterized by a simple set of conditions on the subdifferential of the expected revenue function. These conditions are appropriate for either the discrete or continuous demand cases. These conditions are further simplified to a set of conditions that relate the probability distributions of demand for the various fare classes to their respective fares. The latter conditions are guaranteed to have a solution when the joint probability distribution of demand is continuous. Characterization of the problem as a series of monotone optimal stopping problems proves optimality of the fixed-limit policy over all admissible policies. A comparison is made of the optimal solutions ...

On the Convergence of Policy Iteration in Stationary Dynamic Programming

The policy iteration method of dynamic programming is studied in an abstract setting. It is shown to be equivalent to the Newton-Kantorovich iteration procedure applied to the functional equation of dynamic programming. This equivalence is used to obtain the rate of convergence and error bounds for the sequence of values generated by policy iteration. These results are discussed in the context of the finite state Markovian decision problem with compact action space. An example is analyzed in detail.

Some Inequalities for Parallel-Server Queues

This paper obtains bounds, in terms of the first two moments of the input, on the expected wait in an A/G/k queue with stationary input. To this end, two single-server systems are constructed. The wait in queue for the first single-server system is stochastically larger than the wait in the given multiserver system, and the expected wait in the second single-server system is used to obtain a lower bound on the expected wait in the A/G/k system. The paper also develops, as a consequence of the lower bounds, some results concerning the optimum number of servers, given a fixed work capacity.

Dynamic Airline Revenue Management with Multiple Semi-Markov Demand

When a customer requests a discount fare, the airline must decide whether to sell the seat at the requested discount or to hold the seat in hope that a customer will arrive later who will pay more. We model this situation for a single-leg flight with multiple fare classes and customers who arrive according to a semi-Markov process (possibly nonhomogeneous). These customers can request multiple seats (batch requests) and can be overbooked. Under certain conditions, we show that the value function decreases as departure approaches. If each customer only requests a single seat or if the requests can be partially satisfied, then we show that there are optimal booking curves which decrease as departure approaches. We also provide counterexamples to show that this structural property of the optimal policy need not hold for more general arrival processes if the requests can be for more than one seat and must be accepted or rejected as a whole.

A Generalization of L = λW to Moments of Queue Length and Waiting Times

The well known formula L = λW relates the time-average number in queue to the expected wait in queue of a customer. This paper specializes a more general formula, denoted by H = λG, in order to obtain relations between moments of L and W other than the first. The basic queue considered is G/G/k with stationary input. The special case where the arrival times form a renewal process and the more special case where they are a Poisson process are also discussed.

A Generalization of Erlang's Loss System to State Dependent Arrival and Service Rates

An Erlang Loss System with state dependent arrival and service rates is examined. This model includes Processor Shared Systems and birth-death processes. The state of the system is the number of occupied servers, the time until the next arrival, and the amounts of work remaining for the customers being served. Stationary probability distributions and conditions for their existence are determined for the continuous time process observed at arrivals and departures. The average delay of a customer is also computed.

The repair kit problem revisited

The repair kit problem is concerned with finding an optimal kit of parts and tools to carry for on-site repairs. The choice of a kit involves evaluating two attributes—an annual holding cost and a penalty for failing to complete repairs. We present a unifying approach for the repair kit problem which demonstrates that a monotone sequence of optimal kits exists for several parameterizations of the objective function combining the two attributes. We analyze the structure of the Pareto set of the convex hull of the kits in the attribute space and show the relationship between the extreme points of these Pareto sets and the optimal kits. Decomposition and various monotonicity properties of the repair kit problem yield some computational simplifications in generating optimal kits.

Framework for the analysis of risks in forest management and silvicultural investments.

Abstract Determination of optimal forest-management regimes has been traditionally based on the assumption that the outcome of the management activity is certain. This was the case despite the almost universal recognition that forest-management outcomes are, in fact, uncertain. In this paper, a comprehensive conceptual framework for incorporating risk in forest-management and silviculture investment is developed. The framework distinguishes between decision problems which are structured and those which are unstructured. For the former, the framework focuses upon achieving a match between the decision-problem representation and the degree of knowledge of risk preferences, risk-preference structures and the nature of the risks. For unstructured decisions a new formulation based upon the concept of resilience is developed. The two framework are reconciled through a calculus of risks, benefits and resilience.

A stochastic model allowing interaction among individuals and its behavior for large populations

An important aspect of many societal and institutional processes is the way in which the involved individuals interact. Motivated by work of Conlisk (1976) we formulate a model which allows such interaction among a finite number of individuals. Conditions are established under which our model converges to an approximation suggested by Conlisk as the population becomes infinite.

Semi-Markov information model for revenue management and dynamic pricing

In traditional airline yield management, when a customer requests a discount fare, the airline must decide whether to sell a seat at the requested discount or to hold the seat in hopes that a customer will arrive later who will pay more. In contrast to that, in dynamic pricing models of revenue management, when faced with a request for a seat the airline quotes a price that may or may not be accepted by that customer. In each approach different type of information is available to the seller and, consequently, there is usually a difference between optimal policies and their expected revenues. On the other hand many structural properties of optimal policies are shared. We provide a framework that includes these two types of models by introducing an information variable into the state description of the decision problem.