In situ column generation for a cutting-stock problem

Lee, Jon

doi:10.1016/j.cor.2005.09.007

Cited by 19 publications

(11 citation statements)

References 25 publications

(41 reference statements)

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“…If the solution of (78) does not satisfy the proper column bound constraints:

\sum_{k' \in K_{s_{k} d_{k}} : t_{k'} \geq t_{k}} \frac{t_{k'}}{t_{k}} x_{k'}^{c} \leq {LHS}_{k} \forall k \in K,

we have to consider alternative grooming patterns that do not correspond to the minimum possible reduced cost for a given optical hop configuration. We have developed two methods for solving the wavelength routing configuration pricing problem with traffic bounds (79): a greedy heuristic and an exact method making use of an in situ column generation technique .…”

Section: Implementation Featuresmentioning

confidence: 99%

“…The greedy algorithm is a standard procedure which is presented in . The in situ column generation is a method to generate columns (grooming patterns in our case) directly in the master program (which is defined by the wavelength configuration pricing problem in our case). Note that, for a given

o \in O

, an alternative to

g_{o}^{⋆}

must be considered only if

g_{o}^{⋆}

carries traffic that is involved in some violated proper column constraints (79).…”

Section: Implementation Featuresmentioning

confidence: 99%

“…The pricing problem of wavelength routing configurations that is given in (24) can be formulated in terms of grooming patterns. For each optical hop configuration o ∈ O, consider the optimal grooming pattern g o with minimum reduced costc g = p y g p − k∈K π k x g k only ifc g < 0.…”

Section: Generating Wavelength Configurationsmentioning

confidence: 99%

See 2 more Smart Citations

Reformulation and decomposition approaches for traffic routing in optical networks

2016

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We consider a multilayer network design model arising from a real‐life telecommunication application where traffic routing decisions imply the installation of expensive nodal equipment. Customer requests come in the form of bandwidth reservations for a given origin destination pair. Bandwidth demands are expressed as multiples of nominal granularities. Each request must be single‐path routed. Grooming several requests on the same wavelength and multiplexing wavelengths in the same optical stream allow a more efficient use of network capacity. However, each addition or withdrawal of a request from a wavelength requires optical to electrical conversion and the use of cross‐connect equipment with expensive ports of high densities. The objective is to minimize the number of required ports of the cross‐connect equipment. We deal with backbone optical networks, therefore with networks with a moderate number of nodes (14 to 20) but thousands of requests. Further difficulties arise from the symmetries in wavelength assignment and traffic loading. Traditional multicommodity network flow approaches are not suited for this problem. Instead, four alternative models relying on Dantzig–Wolfe and/or Benders' decomposition are introduced and compared. The formulations are strengthened using symmetry breaking restrictions, variable domain reduction, zero‐one discretization of integer variables, and cutting planes. The resulting dual bounds are compared to the values of primal solutions obtained through hierarchical optimization and rounding procedures. For realistic size instances, our best approaches provide solutions with optimality gap of approximately 5% on average in around 2 h of computing time. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 67(4), 277–298 2016

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“…If the solution of (78) does not satisfy the proper column bound constraints:

\sum_{k' \in K_{s_{k} d_{k}} : t_{k'} \geq t_{k}} \frac{t_{k'}}{t_{k}} x_{k'}^{c} \leq {LHS}_{k} \forall k \in K,

Section: Implementation Featuresmentioning

confidence: 99%

o \in O

, an alternative to

g_{o}^{⋆}

must be considered only if

g_{o}^{⋆}

carries traffic that is involved in some violated proper column constraints (79).…”

Section: Implementation Featuresmentioning

confidence: 99%

See 1 more Smart Citation

Reformulation and decomposition approaches for traffic routing in optical networks

2016

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“…In addition to the natural mathematical interest in studying mixed-integer quadratically constrained programming, there is a wealth of applications that have motivated the development of practical approaches; for example: Trimloss problems (see [84], for example), portfolio optimization (see [26], for example), Max-Cut and other binary quadratic models (see [108,109] and the references therein).…”

Section: Quadratic Functionsmentioning

confidence: 99%

Nonlinear Integer Programming

Li¹,

Sun²

2006

International Series in Operations Research &Amp; Management Science

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Abstract. Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic.The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms.We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems.It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.

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“…Scholars mainly studied the nesting optimization algorithm, and propose some effective methods and models. There are common optimization methods such as linear programming method (Ariyawansa and Felt 2004;Lee 2007), dynamic programming method (Wolf and Achim 2013;Vanberkel et al 2014), the heuristic algorithm (Ho and Ji 2004;Adzakpa et al 2004;Tuzkaya et al 2013;Terashima-Marin et al 2010;Kiraz et al 2013) and the intelligent optimization algorithm (Tao et al 2014;Fathi and Mozaffari 2014). Chen and Cao (2010) propose a heuristic algorithm for rectangle, which mainly selects the lowest horizontal line rectangle pieces.…”

Section: Introductionmentioning

confidence: 99%

An optimizing model to solve the nesting problem of rectangle pieces based on genetic algorithm

Tang

Guo

et al. 2015

J Intell Manuf

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In the process of cement equipment manufacturing, the demand of rectangle pieces of steel structure is very large. The traditional manual nesting, which is simply cutting by hand-making according to the arrangement of the number and size, causes the low efficiency and material wasting. To solve the problem above, this paper proposes an optimizing model for nesting problem of rectangle pieces. Firstly, with the aim of the maximum utilization ratio of the sheet, the optimization mathematical model for nesting problem of rectangle pieces is established. The lowest horizontal line searching algorithm is described in detail. Secondly, the mathematical model is solved to get the optimal solution by the combination of genetic algorithm and the lowest horizontal line searching algorithm. In the solution process, this paper presents the methods of gene encoding and decoding, definition of fitness function, the design of genetic operators and the design of algorithm operating parameters. Finally, we use one sheet as an example to illustrate the proposed model and algorithm process. Experimental results have shown that B Xixing Li the proposed approach is able to achieve rectangle pieces nesting with the maximum material utilization ratio.

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In situ column generation for a cutting-stock problem

Cited by 19 publications

References 25 publications

Reformulation and decomposition approaches for traffic routing in optical networks

Reformulation and decomposition approaches for traffic routing in optical networks

Nonlinear Integer Programming

An optimizing model to solve the nesting problem of rectangle pieces based on genetic algorithm

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