Branch‐Price‐and‐Cut Algorithms

Desrosiers, Jacques; Lübbecke, Marco E.

doi:10.1002/9780470400531.eorms0118

Cited by 53 publications

(31 citation statements)

References 30 publications

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“…Similarly, when only the number of variables is large, the formulations can be used in column generation or branch-and-price procedures [15]. These procedures can be very effective and so can extensions such as branchand-cut-and-price [48] and branch-and-price for extended formulations [142]. For this reason, when both large and small (usually extended) formulations are available it is not always clear which is more convenient.…”

Section: Large Formulationsmentioning

confidence: 99%

Mixed Integer Linear Programming Formulation Techniques

Vielma¹

2015

SIAM Rev.

268

143

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1. Introduction. Throughout more than 50 years of existence, mixed integer linear programming (MIP) theory and practice have been significantly developed, and it is now an indispensable tool in business and engineering [68,94,104]. Two reasons for the success of MIP are linear programming (LP) based solvers and the modeling flexibility of MIP. We now have several extremely effective state-of-the-art solvers [82,69, 52,171] that incorporate many advanced techniques [1,2,25,23,92,112,24] and, since its early stages, MIP has been used to model a wide range of applications [44,45].While in many cases constructing valid MIP formulations is relatively straightforward, some care should be taken in this construction as certain formulation attributes can significantly reduce the effectiveness of LP-based solvers. Fortunately, constructing formulations that behave well with state-of-the-art solvers can usually be achieved by following simple guidelines described in standard textbooks. However, more advanced techniques can often perform significantly better than textbook formulations and are sometimes a necessity. The main objective of this survey is to summarize the state of the art of such formulation techniques for a wide range of problems. To keep the length of this survey under control, we concentrate on formulations for sets of a mixed integer nature that require both integer constrained and continuous variables. We hence purposefully place less emphasis on some related areas such as combinatorial optimization, quadratic and polynomial 0/1 optimization, and polyhedral approximations of convex sets. These topics are certainly areas of important and active research, so we cover them succinctly in section 12.

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Section: Large Formulationsmentioning

confidence: 99%

Mixed Integer Linear Programming Formulation Techniques

Vielma¹

2015

SIAM Rev.

268

143

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“…That is, the cost of the IRSUSP comes from the rolling stock schedule and no cost stems from the TUSP BAC is a well known technique for efficiently solving large scale MIPs, see e.g. Desrosiers and Lübbecke (2010). This approach combines the addition of cutting planes within a BAB framework.…”

Section: Integrated Approaches For the Irsuspmentioning

confidence: 99%

Integrating rolling stock scheduling with train unit shunting

Haahr¹,

Lusby

2017

European Journal of Operational Research

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Highlights• We develop two integrated frameworks for scheduling rolling stock and train unit shunting.• We provide a comparison of several methods for solving the Track Allocation Problem.• We give a counter example demonstrating the suboptimality of a previously proposed optimal method. AbstractIn this paper we consider integrating two important railway optimization problems, in particular the Rolling Stock Scheduling Problem and the Train Unit Shunting Problem. We present two similar branch-and-cut based approaches to solve this integrated problem and, in addition, provide a comparison of different approaches to solve the so-called Track Assignment problem, a subcomponent of the Train Unit Shunting problem. In this analysis we demonstrate, by way of a counter example, the heuristic nature of a previously argued optimal approach. For the integrated problem we analyze the performance of the proposed approaches on several real-life case studies provided by DSB S-tog, a suburban train operator in the greater Copenhagen area. Computational results confirm the necessity of the integrated approach; high quality solutions to the integrated problem are obtained on instances where a conventional, sequential approach ends in infeasibility. Furthermore, for the considered instances, solutions are typically found within a few minutes, indicating the applicability of the methodology to short-term planning.

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“…The Dantzig-Wolfe decomposition builds on the representation theorems by Minkowski and Weyl (see Schrijver 1986;Desrosiers and Lübbecke 2011) that any vector y N ∈ SP can be reformulated as a convex combination of extreme points plus a nonnegative combination of extreme rays of SP. Moreover, SP is a cone for which the only extreme point is the null vector y N = 0 at zero cost.…”

Section: Dynamic Dantzig-wolfe Decompositionmentioning

confidence: 99%