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.

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Lagrangian Relaxation for Integer Programming

Facility location models for distribution planning

The branch-and-bound procedure for solving mixed integer programming (MIP) problems using linear programming relaxations has been used with great success for decades. Over the years, a variety of researchers have studied ways of making the basic algorithm more effective. Breakthroughs in the fields of computer hardware, computer software, and mathematics have led to increasing success at solving larger and larger MIP instances. The goal of this article is to survey many of the results regarding branch-and-bound search strategies and evaluate them again in light of the other advances that have taken place over the years. In addition, novel search strategies are presented and shown to often perform better than those currently used in practice.

A Computational Study of Search Strategies for Mixed Integer Programming

A unifying framework is developed to facilitate the understanding of most known computational approaches to integer programming. A number of currently operational algorithms are related to this framework, and prospects for future progress are assessed.

Integer Programming Algorithms: A Framework and State-of-the-Art Survey

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.

Lagrangean Relaxation Applied to Capacitated Facility Location Problems

This note proposes two extensions of the successful Beale and Small branch-and-bound mixed-integer algorithm. The integer requirements on nonbasic variables are utilized to calculate stronger "penalties" when searching down the solution tree and to give a stronger criterion for abandoning unprofitable branches of the tree when backtracking. This stronger criterion is obtained by making use of Gomory cutting-plane constraints. These modifications have produced considerable reductions of the searching effort required for pure integer and predominantly integer problems, and have the further advantage of being very easy to incorporate.

J. A. Tomlin

Papers

Technical Note-An Improved Branch-and-Bound Method for Integer Programming