This paper presents an improved mixed-integer programming (MIP) model and effective solution strategies for the facility layout problem and is motivated by the work of Meller et al. (1999). This class of problems seeks to determine a least-cost layout of departments having various size and area requirements within a rectangular building, and it is challenging even for small instances. The difficulty arises from the disjunctive constraints that prevent departmental overlaps and the nonlinear area constraints for each department, which existing models have failed to approximate with adequate accuracy. We develop several modeling and algorithmic enhancements that are demonstrated to produce more accurate solutions while also decreasing the solution effort required. We begin by deriving a novel polyhedral outer approximation scheme that can provide as accurate a representation of the area requirements as desired. We also design alternative methods for reducing problem symmetry, evaluate the performance of several classes of valid inequalities, explore the construction of partial convex hull representations for the disjunctive constraints, and investigate judicious branching variable selection priority schemes. The results indicate a substantial increase in the accuracy of the layout produced, while at the same time providing a dramatic reduction in computational effort. In particular, three previously unsolved test problems from the literature for which Meller et al.'s algorithm terminated prematurely after 24 cpu hours of computation (on a SUN Ultra 2 workstation with 390 MB RAM) with respective optimality gaps of 10.14%, 26.45%, and 40%, have been solved to exact optimality with reasonable effort using our proposed approach.
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The National Collegiate Athletic Association Men's Basketball Tournament is a 65‐team championship in American college basketball, in which a single team is eliminated in a play‐in game, followed by a six‐round, single‐elimination tournament. Owing to its immense popularity, each aspect of the tournament, including the selection of teams, their ranking (or “seeding”), and the assignment of teams to locations for the first few rounds, is a hotly debated topic. In this paper, we concentrate on the latter aspect of the tournament composition, as the first two elements have been extensively researched in a variety of settings. We formulate the team assignment problem as a mixed‐integer program, and examine various methods of tightening the linear programming relaxation in order to yield an effective methodology for generating feasible tournament pairings with minimum expected travel. Finally, we demonstrate the results of our algorithm on data derived from the 2004 and 2005 basketball tournaments.
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