Kumar Muthuraman

A stochastic overbooking model for outpatient clinical scheduling with no-shows

In this paper a stochastic overbooking model is formulated and an appointment scheduling policy is developed for outpatient clinics. The schedule is constructed for a single service period partitioned into time slots of equal length. A clinic scheduler assigns patients to slots through a sequential patient call-in process where the scheduler must provide each calling patient with an appointment time before the patients call terminates. Once an appointment is added to the schedule, it cannot be changed. Each calling patient has a no-show probability, and overbooking is used to compensate for patient no-shows. The scheduling objective captures patient waiting time, staff overtime and patient revenue. Conditions under which the objective evolution is unimodal are derived and the behavior of the scheduling policy is investigated under a variety of conditions. Practical observations on the performance of the policy are presented.

Sequential clinical scheduling with patient no-shows and general service time distributions

A sequential clinical scheduling method for patients with general service time distributions is developed in this paper. Patients call a medical clinic to request an appointment with their physician. During the call, the scheduler assigns the patient to an available slot in the physicians schedule. This is communicated to the patient before the call terminates and, thus, the schedule is constructed sequentially. In practice, there is very limited opportunity to adjust the schedule once the complete set of patients is known. Scheduled patients might not attend, that is, they might “no-show,” and the service times of those attending are random. A myopic scheduling algorithm with an optimal stopping criteria for this problem assuming exponential service times already exists in the literature. This work relaxes this assumption and develops numerical techniques for general service time distributions. A special case in which service times are gamma distributed is considered and it is shown that computation is significantly reduced. Finally, exhaustive experimental results are provided along with discussions that provide insights into the practical aspects of the scheduling approach.

Multidimensional Portfolio Optimization with Proportional Transaction Costs

We provide a computational study of the problem of optimally allocating wealth among multiple stocks and a bank account, to maximize the infinite horizon discounted utility of consumption. We consider the situation where the transfer of wealth from one asset to another involves transaction costs that are proportional to the amount of wealth transferred. Our model allows for correlation between the price processes, which in turn gives rise to interesting hedging strategies. This results in a stochastic control problem with both drift-rate and singular controls, which can be recast as a free boundary problem in partial differential equations. Adapting the finite element method and using an iterative procedure that converts the free boundary problem into a sequence of fixed boundary problems, we provide an efficient numerical method for solving this problem. We present computational results that describe the impact of volatility, risk aversion of the investor, level of transaction costs, and correlation among the risky assets on the structure of the optimal policy. Finally we suggest and quantify some heuristic approximations.

A Numerical Method for Solving Singular Stochastic Control Problems

Singular stochastic control has found diverse applications in operations management, economics, and finance. However, in all but the simplest of cases, singular stochastic control problems cannot be solved analytically. In this paper, we propose a method for numerically solving a class of singular stochastic control problems. We combine finite element methods that numerically solve partial differential equations with a policy update procedure based on the principle of smooth pasting to iteratively solve Hamilton-Jacobi-Bellman equations associated with the stochastic control problem. A key feature of our method is that the presence of singular controls simplifies the procedure. We illustrate the method on two examples of singular stochastic control problems, one drawn from economics and the other from queueing systems.

Optimal and approximate algorithms for sequential clinical scheduling with no-shows

The accessibility and efficiency of outpatient clinic operations are largely affected by appointment schedules. Clinical scheduling is a process of assigning physician appointment times to sequentially calling patients. A significant problem in clinical operations is patient no-show, that is, scheduled patients not showing for their appointments. Overbooking can compensate revenue loss due to no-show, but naive overbooking can result in longer patient waiting times and uneven physician work loads. In the past few years, new overbooking methods have been developed for sequential scheduling that yield higher expected profit than simple scheduling rules, but these often fail to exploit information about the future call-in process (they are myopic). To fully use this important information, we develop a Markov Decision Processes (MDP) model for sequential clinical scheduling that books patients to optimize the performance of clinic operations. The model is solved by Dynamic Programming (DP) for small problems. Approximate Dynamic Programming (ADP) algorithms based on aggregation and simulation are developed to find schedules for larger problems. Our computational experiments indicate good improvement over myopic methods.

Simulation Based Portfolio Optimization for Large Portfolios with Transaction Costs

We consider a portfolio optimization problem where the investors objective is to maximize the long-term expected growth rate, in the presence of proportional transaction costs. This problem belongs to the class of stochastic control problems with singular controls, which are usually solved by computing solutions to related partial differential equations called the free-boundary Hamilton Jacobi Bellman (HJB) equations. The dimensionality of the HJB equals the number of stocks in the portfolio. The runtime of existing solution methods grow super-exponentially with dimension, making them unsuitable to compute optimal solutions to portfolio optimization problems with even four stocks. In this work we first present a boundary update procedure that converts the free boundary problem into a sequence of fixed boundary problems. Then by combining simulation with the boundary update procedure, we provide a computational scheme whose runtime, as shown by the numerical tests, scales polynomially in dimension. The results are compared and corroborated against existing methods that scale super-exponentially in dimension. The method presented herein enables the first ever computational solution to free-boundary problems in dimensions greater than three.

Inventory Management with Stochastic Lead Times

This article analyzes a continuous time back-ordered inventory system with stochastic demand and stochastic delivery lags for placed orders. This problem in general has an infinite dimensional state space and is hence intractable. We first obtain the set of minimal conditions for reducing such a systems state space to one dimension and show how this reduction is done. Next, by modeling demand as a diffusion process, we reformulate the inventory control problem as an impulse control problem. We simplify the impulse control problem to a Quasi-Variation Inequality (QVI). Based on the QVI formulation, we obtain the optimality of the (s, S) policy and the limiting distribution of the inventory level. We also obtain the long run average cost of such an inventory system. Finally, we provide a method to solve the QVI formulation. Using a set of computational experiments, we show that significant losses are incurred in approximating a stochastic lead-time system with a fixed lead-time system, thereby highlighting...

American Options Under Stochastic Volatility

The problem of pricing an American option written on an underlying asset with constant price volatility has been studied extensively in literature. Real-world data, however, demonstrate that volatility is not constant, and stochastic volatility models are used to account for dynamic volatility changes. Option pricing methods that have been developed in literature for pricing under stochastic volatility focus mostly on European options. We consider the problem of pricing American options under stochastic volatility, which has had relatively much less attention from literature. First, we develop a transformation procedure to compute the optimal-exercise policy and option price and provide theoretical guarantees for convergence. Second, using this computational tool, we explore a variety of questions that seek insights into the dependence of option prices, exercise policies, and implied volatilities on the market price of volatility risk and correlation between the asset and stochastic volatility. The speed and accuracy of the procedure are compared against existing methods as well.

Regulation of Natural Gas Distribution Using Policy Benchmarks

Local distribution companies (LDCs) play the role of purchasing and delivering natural gas to their consumers, and state regulators oversee the pricing of natural gas to consumers. The common method of regulation, based on the cost of service, provides arguably little incentive for the LDC to optimally manage their procurement activities. In the light of recent deregulation and other changes, benchmarking-based regulatory schemes are being increasingly perceived as the right direction to pursue. Various states are experimenting with simple benchmark mechanisms that have inherent deficiencies and are often criticized. In this paper, we propose and characterize a new kind of benchmark that we call a policy benchmark as a mechanism for regulation. Using variance as the measure of risk, we formulate the regulators and the LDCs problems as multiple-objective optimizations. We provide rigorous characterizations of the dominance frontiers for a two-stage model. We also provide multistage formulations that take into account various natural gas market microstructures. We compute solutions under parameters estimated from relevant real-world data and illustrate that the structures of the dominance frontiers remain unaltered from the characterizations provided by a stylized two-stage model.

An approximate moving boundary method for American option pricing

We present a method to solve the free-boundary problem that arises in the pricing of classical American options. Such free-boundary problems arise when one attempts to solve optimal-stopping problems set in continuous time. American option pricing is one of the most popular optimal-stopping problems considered in literature. The method presented in this paper primarily shows how one can leverage on a one factor approximation and the moving boundary approach to construct a solution mechanism. The result is an algorithm that has superior runtimes-accuracy balance to other computational methods that are available to solve the free-boundary problems. Exhaustive comparisons to other pricing methods are provided. We also discuss a variant of the proposed algorithm that allows for the computation of only one option price rather than the entire price function, when the requirement is such.