418 research outputs found
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Primal-dual variable neighborhood search for the simple plant-location problem
Copyright @ 2007 INFORMSThe variable neighborhood search metaheuristic is applied to the primal simple plant-location problem and to a reduced dual obtained by exploiting the complementary slackness conditions. This leads to (i) heuristic resolution of (metric) instances with uniform fixed costs, up to n = 15,000 users, and m = n potential locations for facilities with an error not exceeding 0.04%; (ii) exact solution of such instances with up to m = n = 7,000; and (iii) exact solutions of instances with variable fixed costs and up to m = n = 15, 000.This work is supported by NSERC Grant 105574-02; NSERC Grant OGP205041; and partly by the Serbian Ministry of Science, Project 1583
Exact algorithms for the order picking problem
Order picking is the problem of collecting a set of products in a warehouse
in a minimum amount of time. It is currently a major bottleneck in supply-chain
because of its cost in time and labor force. This article presents two exact
and effective algorithms for this problem. Firstly, a sparse formulation in
mixed-integer programming is strengthened by preprocessing and valid
inequalities. Secondly, a dynamic programming approach generalizing known
algorithms for two or three cross-aisles is proposed and evaluated
experimentally. Performances of these algorithms are reported and compared with
the Traveling Salesman Problem (TSP) solver Concorde
The Strong Perfect Graph Conjecture: 40 years of Attempts, and its Resolution
International audienceThe Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods. The first of these three approaches yielded the first (and to date only) proof of the SPGC; the other two remain promising to consider in attempting an alternative proof. This paper is an unbalanced survey of the attempts to solve the SPGC; unbalanced, because (1) we devote a signicant part of it to the 'primitive graphs and structural faults' paradigm which led to the Strong Perfect Graph Theorem (SPGT); (2) we briefly present the other "direct" attempts, that is, the ones for which results exist showing one (possible) way to the proof; (3) we ignore entirely the "indirect" approaches whose aim was to get more information about the properties and structure of perfect graphs, without a direct impact on the SPGC. Our aim in this paper is to trace the path that led to the proof of the SPGT as completely as possible. Of course, this implies large overlaps with the recent book on perfect graphs [J.L. Ramirez-Alfonsin and B.A. Reed, eds., Perfect Graphs (Wiley & Sons, 2001).], but it also implies a deeper analysis (with additional results) and another viewpoint on the topic
Branching via Cutting Plane Selection: Improving Hybrid Branching
Cutting planes and branching are two of the most important algorithms for
solving mixed-integer linear programs. For both algorithms, disjunctions play
an important role, being used both as branching candidates and as the
foundation for some cutting planes. We relate branching decisions and cutting
planes to each other through the underlying disjunctions that they are based
on, with a focus on Gomory mixed-integer cuts and their corresponding split
disjunctions. We show that selecting branching decisions based on quality
measures of Gomory mixed-integer cuts leads to relatively small
branch-and-bound trees, and that the result improves when using cuts that more
accurately represent the branching decisions. Finally, we show how the history
of previously computed Gomory mixed-integer cuts can be used to improve the
performance of the state-of-the-art hybrid branching rule of SCIP. Our results
show a 4\% decrease in solve time, and an 8\% decrease in number of nodes over
affected instances of MIPLIB 2017
Relaxations of mixed integer sets from lattice-free polyhedra
This paper gives an introduction to a recently established link between the geometry of numbers and mixed integer optimization. The main focus is to provide a review of families of lattice-free polyhedra and their use in a disjunctive programming approach. The use of lattice-free polyhedra in the context of deriving and explaining cutting planes for mixed integer programs is not only mathematically interesting, but it leads to some fundamental new discoveries, such as an understanding under which conditions cutting planes algorithms converge finitel
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