He is a member of the Operations Research Society of America, the Mathematical Programming Society, and the Society for Industrial and Applied Mathematics. Research Interests Linear programming is concerned with certain kinds of problems related to the optimal allocation of scarce resources among competing, but otherwise independent, activities. For more than forty years linear programming has been an extremely effective tool in widespread use by decision-makers in business and government. In order to develop more efficient algorithms capable of broader applicability, it would be helpful to have a better understanding of linear programming problems. A principal focus of our work has been to provide a precise and concise description of the combinatorial structure underlying linear programming duality theory.
Related work concerns variations on linear programming.
For example, certain production-scheduling problems are complicated by the possibility that items loaded at different times may be scheduled to be simultaneously resident on a machine with fixed capacity. For a particular class of problems, we have shown that the capacity conditions can be incorporated in a surprisingly easy way into an alternative model that is solvable by special-purpose techniques used in network flow theory. The main tool used here is geometric duality of planar graphs. We have also done extensive computational work on network flow codes, employing statistical methods to model how execution times respond to variations in input characteristics. For example, an ongoing project investigates the use of heuristic techniques in the efficient sequencing of x-ray diffraction readings in crystallographic experiments.