Reformulation and Decomposition of Integer Programs

Vanderbeck, François; Wolsey, Laurence A.

doi:10.1007/978-3-540-68279-0_13

Cited by 110 publications

(82 citation statements)

References 87 publications

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“…Downloaded 04/02/15 to 18.51.1.3. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php For more information on extended formulations in combinatorial optimization we refer the reader to [35,154,95] and for more on polyhedral approximations of convex sets we refer the reader to [14,16,152] and [93,Chapters 8 and 17].…”

Section: Combinatorial Optimization and Approximation Of Convexmentioning

confidence: 99%

Mixed Integer Linear Programming Formulation Techniques

Vielma¹

2015

SIAM Rev.

270

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: Combinatorial Optimization and Approximation Of Convexmentioning

confidence: 99%

Mixed Integer Linear Programming Formulation Techniques

Vielma¹

2015

SIAM Rev.

270

143

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“…We first use combo as a heuristic by feeding it with m items j of profitπ j and weight w j (just one item per item type). If this attempt fails in finding a negative reduced cost column, then we use combo as an exact approach by feeding it with the entire set of items, but invoking a binary expansion (see, e.g., Vanderbeck and Wolsey [287] Let LB = ⌈L(F P R )⌉ denote the lower bound that we obtained, P A ⊆ P ′ the set of columns that have been generated to reach linear optimality, and P B ⊆ P A the set of columns that belong to the optimal basis. A classical way to obtain an upper bound from this information is to solve to optimal integrality the restricted master problem with the…”

Section: Relations Among Modelsmentioning

confidence: 99%

“…A seventh, more powerful, formulation is even proposed in this thesis (Chapter 4). Vanderbeck and Wolsey [287] described in details generic procedures to obtain reformulations, and categorized them in several categories. In the extended formulations, new variables are introduced so as to better model the structure When the problem is very complex, which is often the case when C&PP involve two or three dimensions with non overlapping restrictions, MILP models usually struggle to find optimal solutions, as the number of variables and constraints they involve are too high.…”

Section: Introductionmentioning

confidence: 99%

Mathematical models and decomposition algorithms for cutting and packing problems

Delorme¹

2017

4OR-Q J Oper Res

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“…These three smaller problems were solved in phases and each of them was formulated with mathematical programming and solved by an exact solver. For detailed reviews of decomposition approaches see (Ralphs and Galati, 2010;Vanderbeck and Wolsey, 2010).…”

Section: Literature Reviewmentioning

confidence: 99%

Mixed Integer Programming with Decomposition to Solve a Workforce Scheduling and Routing Problem

Laesanklang

Landa-Silva

Castillo-Salazar

2015

Proceedings of the International Conference on Operations Research and Enterprise Systems

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Abstract:We propose an approach based on mixed integer programming (MIP) with decomposition to solve a workforce scheduling and routing problem, in which a set of workers should be assigned to tasks that are distributed across different geographical locations. This problem arises from a number of home care planning scenarios in the UK, faced by our industrial partner. We present a mixed integer programming model that incorporates important real-world features of the problem such as defined geographical regions and flexibility in the workers' availability. Given the size of the real-world instances, we propose to decompose the problem based on geographical areas. We show that the quality of the overall solution is affected by the ordering in which the sub-problems are tackled. Hence, we investigate different ordering strategies to solve the sub-problems and show that such decomposition approach is a very promising technique to produce high-quality solutions in practical computational times using an exact optimization method.

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Reformulation and Decomposition of Integer Programs

Cited by 110 publications

References 87 publications

Mixed Integer Linear Programming Formulation Techniques

Mixed Integer Linear Programming Formulation Techniques

Mathematical models and decomposition algorithms for cutting and packing problems

Mixed Integer Programming with Decomposition to Solve a Workforce Scheduling and Routing Problem

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