John R. Birge

Decomposition and Partitioning Methods for Multistage Stochastic Linear Programs

Multistage stochastic linear programs model problems in financial planning, dynamic traffic assignment, economic policy analysis, and many other applications. Equivalent representations of such problems as deterministic linear programs are, however, excessively large. This paper develops decomposition and partitioning methods for solving these problems and reports on computational results on a set of practical test problems.

A stochastic model for the unit commitment problem

The authors develop a model and a solution technique for the problem of generating electric power when demands are not certain. They also provide techniques for improving the current methods used in solving the traditional unit commitment problem. The solution strategy can be run in parallel due to the separable nature of the relaxation used. Numerical results indicate significant savings in the cost of operating power generating systems when the stochastic model is used instead of the deterministic model.

A Multicut Algorithm for Two-Stage Stochastic Linear Programs.

Outer linearization methods, such as Van Slyke and Wetss L-shaped method for stochastic linear programs, generally apply a single cut on the nonlinear objective at each major iteration. The structure of stochastic programs allows for several cuts to be placed at once. This paper describes a multicut algorithm to carry out this procedure. It presents experimental and theoretical justification for reductions in major iterations.

Matchup Scheduling with Multiple Resources, Release Dates and Disruptions

This paper considers the rescheduling of operations with release dates and multiple resources when disruptions prevent the use of a preplanned schedule. The overall strategy is to follow the preschedule until a disruption occurs. After a disruption, part of the schedule is reconstructed to match up with the preschedule at some future time. Conditions are given for the optimality of this approach. A practical implementation is compared with the alternatives of preplanned static scheduling and myopic dynamic scheduling. A set of practical test problems demonstrates the advantages of the matchup approach. We also explore the solution of the matchup scheduling problem and show the advantages of an integer programming approach for allocating resources to jobs.

The value of the stochastic solution in stochastic linear programs with fixed recourse

Stochastic linear programs have been rarely used in practical situations largely because of their complexity. In evaluating these problems without finding the exact solution, a common method has been to find bounds on the expected value of perfect information. In this paper, we consider a different method. We present bounds on the value of the stochastic solution, that is, the potential benefit from solving the stochastic program over solving a deterministic program in which expected values have replaced random parameters. These bounds are calculated by solving smaller programs related to the stochastic recourse problem.

A Stochastic Programming Approach to the Airline Crew Scheduling Problem

Traditional methods model the billion-dollar airline crew scheduling problem as deterministic and do not explicitly include information on potential disruptions. Instead of modeling the crew scheduling problem as deterministic, we consider a stochastic crew scheduling model and devise a solution methodology for integrating disruptions in the evaluation of crew schedules. The goal is to use that information to find robust solutions that better withstand disruptions. Such an approach is important because we can proactively consider the effects of certain scheduling decisions. By identifying more robust schedules, cascading delay effects are minimized. In this paper we describe our stochastic integer programming model for the airline crew scheduling problem and develop a branching algorithm to identify expensive flight connections and find alternative solutions. The branching algorithm uses the structure of the problem to branch simultaneously on multiple variables without invalidating the optimality of the algorithm. We present computational results demonstrating the effectiveness of our branching algorithm.

Option Methods for Incorporating Risk into Linear Capacity Planning Models

Manufacturing and service operations decisions depend critically on capacity and resource limits. These limits directly affect the risk inherent in those decisions. While risk consideration is well developed in finance through efficient market theory and the capital asset pricing model, operations management models do not generally adopt these principles. One reason for this apparent inconsistency may be that analysis of an operational model does not reveal the level of risk until the model is solved. Using results from option pricing theory, we show that this inconsistency can be avoided in a wide range of planning models. By assuming the availability of market hedges, we show that risk can be incorporated into planning models by adjusting capacity and resource levels. The result resolves some possible inconsistencies between finance and operations and provides a financial basis for many planning problems. We illustrate the proposed approach using a capacity-planning example.

A parallel implementation of the nested decomposition algorithm for multistage stochastic linear programs

Multistage stochastic linear programs can represent a variety of practical decision problems. Solving a multistage stochastic program can be viewed as solving a large tree of linear programs. A common approach for solving these problems is the nested decomposition algorithm, which moves up down the tree by solving nodes and passing information among nodes. The natural independence of subtrees suggests that much of the computational effort of the nested decomposition algorithm can run in parallel across small numbers of fast processors. This paper explores the advantages of such parallel implementations over serial implementations and compares alternative sequencing protocols for parallel processors. Computational experience on a large test set of practical problems with up to 1.5 million constraints and almost 5 million variables suggests that parallel implementations may indeed work well, but they require careful attention to processor load balancing.

Single‐machine scheduling subject to stochastic breakdowns

We provide several examples of one‐machine problems in which the minimization of expected cost subject to stochastic breakdowns of the machine can be successfully attacked analytically. In particular for the weighted flow‐time model, we derive strong bounds on the difference between the optimal static policy and the WSPT policy and discuss an example in which the WSPT policy is not optimal.

Using integer programming to refine Lagrangian-based unit commitment solutions

The authors develop a technique for refining the unit commitment obtained from solving the Lagrangian. Their model is a computer program with nonlinear constraints. It can be solved to optimality using the branch-and-bound technique. Numerical results indicate a significant improvement in the quality of the solution obtained.