A New Lower Bound for the Minimum Linear Arrangement of a Graph

Facility layout problems are an important class of operations research problems that has been studied for several decades. Most variants of facility layout are NP-hard, therefore global optimal solutions are difficult or impossible to compute in reasonable time. Mathematical optimization approaches that guarantee global optimality of solutions or tight bounds on the global optimal value have nevertheless been successfully applied to several variants of facility layout. This review covers three classes of layout problems, namely row layout, unequal-areas layout, and multifloor layout. We summarize the main contributions to the area made using mathematical optimization, mostly mixed integer linear optimization and conic optimization. For each class of problems, we also briefly discuss directions that remain open for future research.

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“…Constraints (11) require that at least two of i, j, k must be in the same row. It also ensures that no more than two rows are used.…”

Section: A Model With O(n 2 ) Binary Variablesmentioning

confidence: 99%

“…The single-row case is known in the literature as the linear arrangement problem, see e.g. [63], [11], [12], [80], [6], and is well known to be NP-hard even if all the pairwise costs are binary [35].…”

Section: The Equidistant Mrflpmentioning

confidence: 99%

Mathematical optimization approaches for facility layout problems: The state-of-the-art and future research directions

Anjos

Vieira

2017

European Journal of Operational Research

160

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“…(see e.g., (Amaral et al, 2008), (Horton, 1997), (Caprara and Gonzalez, 2005), (Caprara et al, 2010), (Petit, 1999), (Hungerlaender and Rendl, 2012), (Schwarz, 2010)). There are various mathematical formulations for the MinLA.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…is feasible, with 0 as the objective value. Hence, it is necessary to strengthen the relaxation by introducing valid inequalities (see (Amaral et al, 2008), (Amaral, 2009), (Amaral and Letchford, 2011), (Horton, 1997), (Caprara et al, 2010), (Schwarz, 2010)). We show in Section 3, how to deal with these two aspects.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…For a given graph, the MinLA consists in arranging the nodes of the graph on a line in such a way to minimize the sum of the distance between the adjacent nodes. In the literature, the MinLA is known under different names such as the optimal linear ordering, the edge sum problem, the minimum-1-sum (see e.g., (Amaral et al, 2008), (Horton, 1997), (Caprara and Gonzalez, 2005), (Caprara et al, 2010), (Petit, 1999), (Hungerlaender and Rendl, 2012), (Schwarz, 2010)).…”

Section: Introductionmentioning

confidence: 99%

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A New Mathematical Model For the Minimum Linear Arrangement Problem

Moeini

Gueye

Loyal

2014

Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems

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This paper addresses a classical combinatorial optimization problem called the Minimum Linear Arrangement (MinLA) Problem. The MinLA problem has numerous applications in different domains of science and engineering. It is known to be NP-hard for general graphs. The objective of this paper is to introduce a new mathematical model and associated theoretical results, including novel rank inequalities. Preliminary computational experiments are reported on some benchmark instances.

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