Proving the performance of a new revenue management system

Revenue management (RM) is a complicated business process that can best be described as control of sales (using prices, restrictions, or capacity), usually using software as a tool to aid decisions. RM software can play a mere informative role, supplying analysts with formatted and summarized data who use it to make control decisions (setting a price or allocating capacity for a price point), or, play a deeper role, automating the decisions process completely, at the other extreme. The RM models and algorithms in the academic literature by and large concentrate on the latter, completely automated, level of functionality. A firm considering using a new RM model or RM system needs to evaluate its performance. Academic papers justify the performance of their models using simulations, where customer booking requests are simulated according to some process and model, and the revenue perfor- mance of the algorithm compared to an alternate set of algorithms. Such simulations, while an accepted part of the academic literature, and indeed providing research insight, often lack credibility with management. Even methodologically, they are usually awed, as the simula- tions only test \within-model" performance, and say nothing as to the appropriateness of the model in the first place. Even simulations that test against alternate models or competition are limited by their inherent necessity on fixing some model as the universe for their testing. These problems are exacerbated with RM models that attempt to model customer purchase behav- ior or competition, as the right models for competitive actions or customer purchases remain somewhat of a mystery, or at least with no consensus on their validity. How then to validate a model? Putting it another way, we want to show that a particular model or algorithm is the cause of a certain improvement to the RM process compared to the existing process. We take care to emphasize that we want to prove the said model as the cause of performance, and to compare against a (incumbent) process rather than against an alternate model. In this paper we describe a \live" testing experiment that we conducted at Iberia Airlines on a set of flights. A set of competing algorithms control a set of flights during adjacent weeks, and their behavior and results are observed over a relatively long period of time (9 months). In parallel, a group of control flights were managed using the traditional mix of manual and algorithmic control (incumbent system). Such \sandbox" testing, while common at many large internet search and e-commerce companies is relatively rare in the revenue management area. Sandbox testing has an undisputable model of customer behavior but the experimental design and analysis of results is less clear. In this paper we describe the philosophy behind the experiment, the organizational challenges, the design and setup of the experiment, and outline the analysis of the results. This paper is a complement to a (more technical) related paper that describes the econometrics and statistical analysis of the results.Revenue management, airlines, sandbox testing,econometric analysis.

Long-Term Stability Estimates and Existence of a Global Attractor in a Finite Element Approximation of the Navier–Stokes Equations with Numerical Subgrid Scale Modeling

Variational multiscale methods lead to stable finite element approximations of the Navier–Stokes equations, dealing with both the indefinite nature of the system (pressure stability) and the velocity stability loss for high Reynolds numbers. These methods enrich the Galerkin formulation with a subgrid component that is modeled. In fact, the effect of the subgrid scale on the captured scales has been proved to dissipate the proper amount of energy needed to approximate the correct energy spectrum. Thus, they also act as effective large-eddy simulation turbulence models and allow one to compute flows without the need to capture all the scales in the system. In this article, we consider a dynamic subgrid model that enforces the subgrid component to be orthogonal to the finite element space in the L2 sense. We analyze the long-term behavior of the algorithm, proving the existence of appropriate absorbing sets and a compact global attractor. The improvements with respect to a finite element Galerkin approximation are the long-term estimates for the subgrid component, which are translated to effective pressure and velocity stability. Thus, the stabilization introduced by the subgrid model into the finite element problem does not deteriorate for infinite time intervals of computation

Long term stability estimates and existence of a global attractor in a finite element approximation of the Navier-Stokes equations with numerical sub-grid scale modeling

Variational multiscale methods lead to stable finite element approximations of the Navier-Stokes equations, both dealing with the indefinite nature of the system (pressure stability) and the velocity stability loss for high Reynolds numbers. These methods enrich the Galerkin formulation with a sub-grid component that is modelled. In fact, the effect of the sub-grid scale on the captured scales has been proved to dissipate the proper amount of energy needed to approximate the correct energy spectrum. Thus, they also act as effective large-eddy simulation turbulence models and allow to compute flows without the need to capture all the scales in the system. In this article, we consider a dynamic sub-grid model that enforces the sub-grid component to be orthogonal to the finite element space in L2 sense.We analyze the long-term behavior of the algorithm, proving the existence of appropriate absorbing sets and a compact global attractor. The improvements with respect to a finite element Galerkin approximation are the long-term estimates for the sub-grid component, that are translated to effective pressure and velocity stability. Thus, the stabilization introduced by the sub-grid model into the finite element problem is not deteriorated for infinite time intervals of computation.Postprint (published version