Anton J. Kleywegt

The Sample Average Approximation Method for Stochastic Discrete Optimization

In this paper we study a Monte Carlo simulation--based approach to stochastic discrete optimization problems. The basic idea of such methods is that a random sample is generated and the expected value function is approximated by the corresponding sample average function. The obtained sample average optimization problem is solved, and the procedure is repeated several times until a stopping criterion is satisfied. We discuss convergence rates, stopping rules, and computational complexity of this procedure and present a numerical example for the stochastic knapsack problem.

The Sample Average Approximation Method Applied to Stochastic Routing Problems: A Computational Study

The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is approximated by a sample average estimate derived from a random sample. The resulting sample average approximating problem is then solved by deterministic optimization techniques. The process is repeated with different samples to obtain candidate solutions along with statistical estimates of their optimality gaps.We present a detailed computational study of the application of the SAA method to solve three classes of stochastic routing problems. These stochastic problems involve an extremely large number of scenarios and first-stage integer variables. For each of the three problem classes, we use decomposition and branch-and-cut to solve the approximating problem within the SAA scheme. Our computational results indicate that the proposed method is successful in solving problems with up to 21694 scenarios to within an estimated 1.0% of optimality. Furthermore, a surprising observation is that the number of optimality cuts required to solve the approximating problem to optimality does not significantly increase with the size of the sample. Therefore, the observed computation times needed to find optimal solutions to the approximating problems grow only linearly with the sample size. As a result, we are able to find provably near-optimal solutions to these difficult stochastic programs using only a moderate amount of computation time.

Auction-Based Multi-Robot Routing.

Recently, auction methods have been investigated as effective, decentralized methods for multi-robot coordination. Experimental research has shown great potential, but has not been complemented yet by theoretical analysis. In this paper we contribute a theoretical analysis of the performance of auction methods for multi-robot routing. We suggest a generic framework for auction-based multi-robot routing and analyze a variety of bidding rules for different team objectives. This is the first time that auction methods are shown to offer theoretical guarantees for such a variety of bidding rules and team objectives.

The Inventory Routing Problem

The role of logistics management is changing. Many companies are realizing that value for a customer can, in part, be created through logistics management (Langley and Holcomb, 1996). Customer value can be created through product availability, timeliness and consistency of delivery, ease of placing orders, and other elements of logistics service. Consequently, logistics service is becoming recognized as an essential element of customer satisfaction in a growing number of product markets today.

The Dynamic and Stochastic Knapsack Problem

The Dynamic and Stochastic Knapsack Problem (DSKP) is defined as follows. Items arrive according to a Poisson process in time. Each item has a demand (size) for a limited resource (the knapsack) and an associated reward. The resource requirements and rewards are jointly distributed according to a known probability distribution and become known at the time of the items arrival. Items can be either accepted or rejected. If an item is accepted, the items reward is received; and if an item is rejected, a penalty is paid. The problem can be stopped at any time, at which time a terminal value is received, which may depend on the amount of resource remaining. Given the waiting cost and the time horizon of the problem, the objective is to determine the optim al policy that maximizes the expected value (rewards minus costs) accumulated. Assuming that all items have equal sizes but random rewards, optimal solutions are derived for a variety of cost structures and time horizons, and recursive algorithms for computing them are developed. Optimal closed-form solutions are obtained for special cases. The DSKP has applications in freight transportation, in scheduling of batch processors, in selling of assets, and in selection of investment projects.>

The Dynamic and Stochastic Knapsack Problem with Random Sized Items

A resource allocation problem, called the dynamic and stochastic knapsack problem (DSKP), is studied. A known quantity of resource is available, and demands for the resource arrive randomly over time. Each demand requires an amount of resource and has an associated reward. The resource requirements and rewards are unknown before arrival and become known at the time of the demands arrival. Demands can be either accepted or rejected. If a demand is accepted, the associated reward is received; if a demand is rejected, a penalty is incurred. The problem can be stopped at any time, at which time a terminal value is received that depends on the quantity of resource remaining. A holding cost that depends on the amount of resource allocated is incurred until the process is stopped. The objective is to determine an optimal policy for accepting demands and for stopping that maximizes the expected value (rewards minus costs) accumulated. The DSKP is analyzed for both the infinite horizon and the finite horizon cases. It is shown that the DSKP has an optimal policy that consists of an easily computed threshold acceptance rule and an optimal stopping rule. A number of monotonicity and convexity properties are studied. This problem is motivated by the issues facing a manager of an LTL transportation operation regarding the acceptance of loads and the dispatching of a vehicle. It also has applications in many other areas, such as the scheduling of batch processors, the selling of assets, the selection of investment projects, and yield management.

Airline Crew Scheduling Under Uncertainty

Airline crew scheduling algorithms widely used in practice assume no disruptions. Because disruptions often occur, the actual cost of the resulting crew schedules is often greater. We consider algorithms for finding crew schedules that perform well in practice. The deterministic crew scheduling model is an approximation of crew scheduling under uncertainty with the assumption that all pairings will operate as planned. We seek better approximate solution methods for crew scheduling under uncertainty that still remain tractable. We give computational results from three fleets that indicate that the crew schedules obtained from our method perform better in a model of operations with disruptions than the crew schedules found via deterministic methods. Under mild assumptions we provide a lower bound on the cost of an optimal crew schedule in operations, and we demonstrate that some of the crew schedules found using our method perform very well relative to this lower bound.

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.

MINIMAX ANALYSIS OF STOCHASTIC PROBLEMS

In practical applications of stochastic programming the involved probability distributions are never known exactly. One can try to hedge against the worst expected value resulting from a considered set of permissible distributions. This leads to a min-max formulation of the corresponding stochastic programming problem. We show that, under mild regularity conditions, such a min-max problem generates a probability distribution on the set of permissible distributions with the min-max problem being equivalent to the expected value problem with respect to the corresponding weighted distribution. We consider examples of the news vendor problem, the problem of moments and problems involving unimodal distributions. Finally, we discuss the Monte Carlo sample average approach to solving such min-max problems.

A Stochastic Model of Airline Operations

We present a stochastic model of the daily operations of an airline. Its primary purpose is to evaluate plans, such as crew schedules, as well as recovery policies in a random environment. We describe the structure of the stochastic model, sources of disruptions, recovery policies, and performance measures. Then, we describe SimAir--our simulation implementation of the stochastic model, and we give computational results. Finally, we give future directions for the study of airline recovery policies and planning under uncertainty.