Arthur M. Geoffrion

Lagrangian Relaxation for Integer Programming

It is a pleasure to write this commentary because it offers an opportunity to express my gratitude to several people who helped me in ways that turned out to be essential to the birth of [8]. They also had a good deal to do with shaping my early career and, consequently, much of what followed.

Lagrangean Relaxation Applied to Capacitated Facility Location Problems

Abstract Lagrangean relaxation, a technique of quite general applicability, is studied in the particular context of the capacitated facility location problem with arbitrary additional constraints. For this class of problems we are able to obtain a reasonably complete algebraic and geometric understanding of how and why Lagrangean relaxation works. Extensive computational results are also reported. Although this work finds immediate application to improved computational procedures for the class of problems studied, our longer term aim is to encourage similar in-depth studies of Lagrangean relaxation for other important classes of problems.

The formal aspects of structured modeling

Structured modeling is an approach to the development of a new generation of computer-based modeling environments. This paper, which is part of a series, presents a formal development of the definitions and theory of structured modeling.

Scheduling Parallel Production Lines with Changeover Costs: Practical Application of a Quadratic Assignment/LP Approach

Production orders for a number of products must be scheduled on a number of similar production lines so as to minimize the sum of product-dependent changeover costs, production costs, and time-constraint penalties. We treat the problem by a quadratic assignment algorithm with a linear programming adjustment, and describe a successful practical application for chemical reactor scheduling.

Primal Resource-Directive Approaches for Optimizing Nonlinear Decomposable Systems

This study presents some new results on three primal-feasible computational approaches for optimizing a system composed of interrelated subsystems. The general structure treated is the same as the principal one of the classic paper by Dantzig and Wolfe, except that convex nonlinearities are permitted, provided that the overall criterion function and coupling constraints are separable by subsystem. Each approach decentralizes the optimization by iteratively allocating system resources to the subsystems, with each subsystem computing its own optimal utilization of the given resources at each iteration. The chief obstacle to directing the resource allocation centrally toward an overall optimum is that the optimal response of each subsystem, as a function of its allowed resources, is not available explicitly. All three procedures therefore approximate or generate the optimal response functions “as needed.”

Prospects for Operations Research in the E-Business Era

The digital economy is creating abundant opportunities for operations research (OR) applications. Several factions of the profession are beginning to respond aggressively, leading to notable successes in such areas as financial services, electronic markets, network infrastructure, packaged OR-software tools, supply-chain management, and travel-related services. Because OR is well matched to the needs of the digital economy in certain ways and because certain enabling conditions are coming to pass, prospects are good for OR to team with related analytic technologies and join information technology as a vital engine of further development for the digital economy. OR professionals should prepare for a future in which most businesses will be e-businesses.

Computer-based modeling environments

This paper gives the authors views on the kind of computer-based modeling environment needed to properly support management science/operations research work, and on the design challenges that need to be met in order to bring such modeling environments into being. It is a written version of the main ideas of two addresses: a plenary at IFORS 87 in Buenos Aires (August, 1987), and the keynote at the 1988 Canadian Operations Research Society Meeting in Montreal (May, 1988).

The SML Language for Structured Modeling: Levels 3 and 4

This is the second of two articles on the principal features of SML, a language for expressing structured models. The prior article covered levels 1 and 2. The present article covers the remaining levels, with special attention to the characteristics of SML that, collectively, make it unique. The intended audience includes evaluators of other modeling languages, designers of modeling languages and systems, and those following the development of structured modeling.

Forces, trends and opportunities in MS/OR

The purposes of this paper—a revised and extended version of the Omega Rho Lecture given at the November 1991 ORSA/TIMS Joint National Meeting—are to assess some important aspects of the current MS/OR situation and to draw some conclusions about desirable future emphases. To these ends, it identifies and discusses four forces of historic importance (the microcomputer and communications revolutions, the dispersion of MS/OR in industry, and academias unbalanced reward structure), three major trends (rapidly disseminating MS/OR tools, declining enrollments of native-born students, and persisting management apathy toward MS/OR), and five outstanding opportunities (ride the computer and communications revolutions, support dispersed practitioners, focus on the service sector, stress embedded applications, and go into the quality business). An underlying theme is that the field will flourish in proportion to how astutely individuals, organizations, professional societies, and universities adapt to the changing ...

OBJECTIVE FUNCTION APPROXIMATIONS IN MATHEMATICAL PROGRAMMING

Mathematical programming applications often require an objective function to be approximated by one of simpler form so that an available computational approach can be used. An a priori bound is derived on the amount of error (suitably defined) which such an approximation can induce. This leads to a natural criterion for selecting the “best” approximation from any given class. We show that this criterion is equivalent for all practical purposes to the familiar Chebyshev approximation criterion. This gains access to the rich legacy on Chebyshev approximation techniques, to which we add some new methods for cases of particular interest in mathematical programming. Some results relating to post-computational bounds are also obtained.