William L. Cooper

Models of the Spiral-Down Effect in Revenue Management

The spiral-down effect occurs when incorrect assumptions about customer behavior cause high-fare ticket sales, protection levels, and revenues to systematically decrease over time. If an airline decides how many seats to protect for sale at a high fare based on past high-fare sales, while neglecting to account for the fact that availability of low-fare tickets will reduce high-fare sales, then high-fare sales will decrease, resulting in lower future estimates of high-fare demand. This subsequently yields lower protection levels for high-fare tickets, greater availability of low-fare tickets, and even lower high-fare ticket sales. The pattern continues, resulting in a so-called spiral down. We develop a mathematical framework to analyze the process by which airlines forecast demand and optimize booking controls over a sequence of flights. Within the framework, we give conditions under which spiral down occurs.

Asymptotic Behavior of an Allocation Policy for Revenue Management

Revenue management has become an important tool in the airline, hotel, and rental car industries. We describe asymptotic properties of revenue management policies derived from the solution of a deterministic optimization problem. Our primary results state that, within a stochastic and dynamic framework, solutions arising out of a single well-known linear program can be used to generate allocation policies for which the normalized revenue converges in distribution to a constant upper bound on the optimal value. We also show similar asymptotic results for expected revenues. In addition, we describe counterintuitive behavior that can occur when allocations are updated during the booking process (updating allocations can lead to lower expected revenue). These results add to the understanding of allocation policies and help to make concrete the statement that simple policies from easy-to-solve formulations can be relatively effective, even when analyzed in the more realistic stochastic and dynamic framework.

Pricing substitutable flights in airline revenue management

We develop a Markov decision process formulation of a dynamic pricing problem for multiple substitutable flights between the same origin and destination, taking into account customer choice among the flights. The model is rendered computationally intractable for exact solution by its multi-dimensional state and action spaces, so we develop and analyze various bounds and heuristics. We first describe three related models, each based on some form of pooling, and introduce heuristics suggested by these models. We also develop separable bounds for the value function which are used to construct value- and policy-approximation heuristics. Extensive numerical experiments show the value- and policy-approximation approaches to work well across a wide range of problem parameters, and to outperform the pooling-based heuristics in most cases. The methods are applicable even for large problems, and are potentially useful for practical applications.

Single-Leg Air-Cargo Revenue Management

We consider a cargo booking problem on a single-leg flight with the goal of maximizing expected contribution. Each piece of cargo is endowed with a random volume and a random weight whose precise values are not known until just before the flights departure. We formulate the problem as a Markov decision process (MDP). Exact solution of the formulation is impractical, because of its high-dimensional state space; therefore, we develop six heuristics. The first four heuristics are based on different value-function approximations derived from two computationally tractable MDPs, each with a one-dimensional state space. The remaining two heuristics are obtained from solving related methematical programming problems. We also compare the heuristics with the first-come, first-served (FCFS) policy. Simulation experiments suggest that the value function approximation derived from separate “volume” and “weight” problems offers the best approach. Comparisons of the expected contribution under the heuristic to an upper bound show that the heuristic is typically close to optimal.

On the Benefits of Pooling in Production-Inventory Systems

We study inventory pooling in systems with symmetric costs where supply lead times are endogenously generated by a finite-capacity production system. We investigate the sensitivity of the cost advantage of inventory pooling to various system parameters, including loading, service levels, demand and production time variability, and structure of the production system. The analysis reveals differences in how various parameters affect the cost reduction from pooling and suggests that these differences stem from the manner in which the parameters influence the induced correlation between lead-time demands of the demand streams. We compare these results with those obtained for pure inventory systems, where lead times are exogenous. We also compare inventory pooling with several forms of capacity pooling.

Stochastic Comparisons in Production Yield Management

Manufacturing firms routinely commit resources to increase yield rates through product- and process-improvement initiatives. Champions of such yield-improvement projects may assume that stochastically larger yield rates are beneficial. In this note, we show that this need not hold, even when the contingent production lot sizes are chosen optimally. We employ stochastic comparison techniques to show that a yield rate that is smaller in the convex order ensures higher expected profit, and we provide a distribution-free bound on the size of increase in expected profit. We also identify properties of yield-rate distributions that do make stochastically larger yield rates beneficial.

Pathwise Properties and Performance Bounds for a Perishable Inventory System

We study a perishable inventory system under a fixed-critical number order policy. By using an appropriate transformation of the state vector, we derive several key sample-path relations. We obtain bounds on the limiting distribution of the number of outdates in a period, and we derive families of upper and lower bounds for the long-run number of outdates per unit time. Analysis of the bounds on the expected number of outdates shows that at least one of the new lower bounds is always greater than or equal to previously published lower bounds, whereas the new upper bounds are sometimes lower than and sometimes higher than the existing upper bounds. In addition, using an expected cost criterion, we compare optimal policies and different choices of critical-number policies.

Stochastic Comparisons in Airline Revenue Management

Consider two markets of different sizes but similar costs and fare structure. All other things being equal, is an airlines expected revenue larger in the market with larger demand? If not, under what circumstances is it possible to compare expected revenues without carrying out a detailed analysis? In this article, we provide answers to these questions by studying the relationship between the optimal expected revenue and the demand distributions when the latter are comparable according to various stochastic orders. For the two-fare class problem with dependent demand we obtain three results. We show that airlines should prefer lesser positive dependence between fare classes when marginal demand distributions are the same. We also describe particular dependence structures under which stochastically larger marginal demand distributions improve optimal expected revenue. Finally, when the dependence between effective demands in the two fare classes arises due to “sell ups,” we show that stochastically larger marginal demand distributions should be preferred. (Sell ups occur when some lower-fare-class customers buy higher-fare tickets upon finding that the former tickets are sold out.) For a problem with an arbitrary number of fare classes and independent demands, we show that stochastically larger demand distributions should be preferred. Numerical examples demonstrating the effect of parameterized demand distributions (with appropriate stochastic ordering) and dependence structures are also presented.

CONVERGENCE OF SIMULATION-BASED POLICY ITERATION

Simulation-based policy iteration (SBPI) is a modification of the policy iteration algorithm for computing optimal policies for Markov decision processes. At each iteration, rather than solving the average evaluation equations, SBPI employs simulation to estimate a solution to these equations. For recurrent average-reward Markov decision processes with finite state and action spaces, we provide easily verifiable conditions that ensure that simulation-based policy iteration almost-surely eventually never leaves the set of optimal decision rules. We analyze three simulation estimators for solutions to the average evaluation equations. Using our general results, we derive simple conditions on the simulation run lengths that guarantee the almost-sure convergence of the algorithm.

Skorohod–Loynes Characterizations of Queueing, Fluid, and Inventory Processes

We consider queueing, fluid and inventory processes whose dynamics are determined by general point processes or random measures that represent inputs and outputs. The state of such a process (the queue length or inventory level) is regulated to stay in a finite or infinite interval – inputs or outputs are disregarded when they would lead to a state outside the interval. The sample paths of the process satisfy an integral equation; the paths have finite local variation and may have discontinuities. We establish the existence and uniqueness of the process based on a Skorohod equation. This leads to an explicit expression for the process on the doubly-infinite time axis. The expression is especially tractable when the process is stationary with stationary input–output measures. This representation is an extension of the classical Loynes representation of stationary waiting times in single-server queues with stationary inputs and services. We also describe several properties of stationary processes: Palm probabilities of the processes at jump times, Little laws for waiting times in the system, finiteness of moments and extensions to tandem and treelike networks.