**Bike Sharing Logistics**

I am part of a team, led by David Shmoys, analyzing bike sharing logistics in partnership with CitiBike in NYC. Our goal is to help CitiBike and other bike-sharing companies with their operational and strategic questions, including how to achieve smooth functioning through a mix of rebalancing, valets, and depots, where and how much to expand their network of stations, and how to design incentive schemes to help balance the flows of bikes in time. The work involves a mix of stochastic modeling and convex optimization.**Structured Simulation Optimization**

The problem of simulation optimization is essentially that of optimizing a function that can only be computed (estimated) through simulation. We have developed techniques to (numerically) detect convexity in a function evaluated only through simulation. Moreover, we have developed a library of simulation-optimization test problems. Applications: Ambulance deployment, call center staffing, control of stochastic processing networks, radiation treatment planning. See here for some TV coverage on ambulances that is related.**Parallel Simulation Optimization**

Parallel computing is ubiquitous today, from laptops to cloud computing to high-performance computing. Each of these environments differs in important ways, but we should be able to exploit their parallel natures in developing simulation optimization algorithms. We are developing parallel ranking and selection algorithms that work in such environments and scale to very large problems. We are also exploring the interplay between search algorithms and ranking and selection algorithms.**Stochastic Root Finding**

This is the problem of identifying a point at which a function equals zero, i.e., a root, when the function or its gradient (in the case it is differentiable) can only be observed with noise. This problem is closely related to simulation optimization, in the sense that the unconstrained minimum of a differentiable function is a root of the gradient. We are developing theory and algorithms for a Bayesian algorithm that updates a prior belief on the location of the root.**Conjoint Analysis**

I am part of a team exploring methods for learning a user's preferences over a large set of items, for use in, e.g., recommendation systems. The algorithm updates a Bayesian prior as it learns more about a user's preferences.

**Dynamic Relocation of Ambulances**

When an ambulance is dispatched to a call it leaves a "hole." Should we try to fill that hole with an available ambulance? What moves should be considered? How can this be done without overly frustrating ambulance crews? We are tackling this question using approximate dynamic programming and simulation. Click here for some TV coverage.**What are the tradeoffs in ambulance fleet design?**

Ambulances can be roughly categorized into two types: Advanced Life Support (ALS) that are staffed by paramedics and deliver the highest level of care, and Basic Life Support (BLS) that deliver high-quality care, but not quite at the same level as paramedics. Some ambulance fleets consist entirely of ALS units, while others are a mix. Why? What are the tradeoffs from an operational perspective?**Statistical Analysis of Emergency Services Data**

The Operations Research models we are applying to try to help Emergency Medical Service (EMS) organizations require a number of input parameters, including call arrival rates in space and time, and travel speeds/times on road networks. We are applying advanced statistical methods to turn Computer-Aided Dispatch data and Automatic Vehicle Location data into reliable estimates of these quantities.**The Game of Monopoly**

The game of Monopoly exhibits many complexities that are faced by real organizations. Specifically, a player has to make decisions in real time in the face of competition from other players and considerable uncertainty about the future. Luck plays a role, but so does strategy. Click here for information on our efforts to understand the game and identify effective strategies, and here for some press coverage.**Low Rank Approximations in Optimization**

Representing complex constraints in high dimensions can require more storage than is currently possible. If one replaces complex constraints with simpler versions, then what is the impact on the optimization problem? This work was motivated through our work in radiation treatment planning where ellipsoidal constraints in 1000 dimensions were replaced with infinitely long cylinders, which could be stored in less than 1% of the memory needed for the ellipsoids.**American Option Pricing and Stochastic Root Finding**

The problem of American option pricing includes a series of decisions: should I stop and exercise the option, or continue? In one dimension, these decisions come down to solving a stochastic root finding problem. We are working on methods to solve such root finding problems efficiently, and to determine how one should allocate computational effort across the different stages in the process to get as accurate an option price as possible.**Variance Reduction Techniques**

Exploring the use of general variance reduction techniques. In particular, looking at the use of martingales to obtain variance reduction in simulations of Markov processes. Current work involves exploring adaptive methods to tune the variance reduction and extending the methods from the Markov setting to general discrete-event simulations.**The Regenerative Method**

The regenerative method of simulation output analysis possesses qualities that make it preferable to other time-average variance constant estimation methods such as batch means. Therefore, it is of great interest to determine how to apply the method to general discrete-event simulations.**Dependence Structures**

The primitive inputs to stochastic models are often assumed to be independent, even when they are known to be dependent in some way. This assumption is usually made to avoid the difficulties of modeling and generating dependent random variables. This work develops methods for modeling and generating dependent random variables.**Input Uncertainty**

There is invariably some uncertainty about the "correct" values of input parameters for simulations, e.g., what is the "correct" arrival rate to a queue? Forecast errors are an example of input uncertainty. In this work we attempt to understand and quantify the impact of that uncertainty.