Time-Indexed Formulations for Machine Scheduling Problems: Column Generation

Akker, J.M. van den; Hurkens, Caj Cor; Savelsbergh, Martin

doi:10.1287/ijoc.12.2.111.11896

Cited by 174 publications

(95 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Consider a single machine scheduling problem on a planning horizon T as studied by van den Akker et al [25]. The problem is to schedule the jobs, j ∈ J = {1, .…”

Section: Machine Schedulingmentioning

confidence: 99%

“…where G is the set of "pseudo-schedules": vector z g and scalar c g define the associated solution and cost for a solution g ∈ G. As done in [24,25], reformulation [M] can be solved by column generation. The pricing subproblem [SP] is a shortest path problem: find a sequence of jobs and down-times to be scheduled on the single machine with possible repetition of jobs.…”

Section: Machine Schedulingmentioning

confidence: 99%

“…where constraints (61) are obtained by summing over k constraints H k z k ≥ h k ∀k, as in the Bin Packing example, when aggregating (17)(18)(19) into (23)(24)(25). The aggregate master takes the form: …”

Section: Reformulationsmentioning

confidence: 99%

See 2 more Smart Citations

Column generation for extended formulations

Sadykov

Vanderbeck

2013

EURO Journal on Computational Optimization

View full text Add to dashboard Cite

Working in an extended variable space allows one to develop tighter reformulations for mixed integer programs. However, the size of the extended formulation grows rapidly too large for a direct treatment by a MIP-solver. Then, one can work with inner approximations defined and improved by generating dynamically variables and constraints. When the extended formulation stems from subproblems' reformulations, one can implement column generation for the extended formulation using a Dantzig-Wolfe decomposition paradigm. Pricing subproblem solutions are expressed in the variables of the extended formulation and added to the current restricted version of the extended formulation along with the subproblem constraints that are active for the subproblem solutions. This so-called "column-and-row generation" procedure is revisited here in a unifying presentation that generalizes the column generation algorithm and extends to the case of working with an approximate extended formulation. The interest of the approach is evaluated numerically on machine scheduling, bin packing, generalized assignment, and multi-echelon lot-sizing problems. We compare a direct handling of the extended formulation, a standard column generation approach, and the "column-and-row generation" procedure, highlighting a key benefit of the latter: lifting pricing problem solutions in the space of the extended formulation permits their recombination into new subproblem solutions and results in faster convergence.

show abstract

“…Consider a single machine scheduling problem on a planning horizon T as studied by van den Akker et al [25]. The problem is to schedule the jobs, j ∈ J = {1, .…”

Section: Machine Schedulingmentioning

confidence: 99%

Section: Machine Schedulingmentioning

confidence: 99%

See 1 more Smart Citation

Column generation for extended formulations

Sadykov

Vanderbeck

2013

EURO Journal on Computational Optimization

View full text Add to dashboard Cite

show abstract

“…However, in the late 1990's, several researchers [5,9,23,22,35,36] independently noted that cuts expressed in terms of variables from a suitable original formulation could be dynamically separated, translated and added to the master problem. Those cuts do not change the structure of the pricing subproblem.…”

Section: Introductionmentioning

confidence: 99%

Robust branch-cut-and-price for the Capacitated Minimum Spanning Tree problem over a large extended formulation

et al. 2006

View full text Add to dashboard Cite

This paper presents a robust branch-cut-and-price algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to q-arbs, a structure that arises from a relaxation of the capacitated prize-collecting arborescence problem in order to make it solvable in pseudo-polynomial time. Traditional inequalities over the arc formulation, like Capacity Cuts, are also used. Moreover, a novel feature is introduced in such kind of algorithms. Powerful new cuts expressed over a very large set of variables could be added, without increasing the complexity of the pricing subproblem or the size of the LPs that are actually solved. Computational results on benchmark instances from the OR-Library show very significant improvements over previous algorithms. Several open instances could be solved to optimality.

show abstract

“…However, for some time this was not considered to be practical [8], since the new dual variables corresponding to separated cuts would have the undesirable effect of changing the structure of the pricing subproblem, making it potentially intractable. However, in the late 1990's, some researchers [7,15,26,27,37,38] independently noted that cuts expressed in terms of variables from a suitable original formulation could be dynamically separated, translated and added to the master problem. Those cuts would not change the structure of the pricing subproblem.…”

Section: Introductionmentioning

confidence: 99%

Robust Branch-Cut-and-Price Algorithms for Vehicle Routing Problems

Pessoa

Aragão

Uchôa

Operations Research/Computer Science Interfaces

View full text Add to dashboard Cite

Summary. This article presents techniques for constructing robust Branch-Cutand-Price algorithms on a number of Vehicle Routing Problem variants. The word "robust" stress the effort of controlling the worst-case complexity of the pricing subproblem, keeping it pseudo-polynomial. Besides summarizing older research on the topic, some promising new lines of investigation are also presented, specially the development of new families of cuts over large extended formulations. Computational experiments over benchmark instances from ACVRP, COVRP, CVRP and HFVRP variants are provided.

show abstract

Time-Indexed Formulations for Machine Scheduling Problems: Column Generation

Cited by 174 publications

References 12 publications

Column generation for extended formulations

Column generation for extended formulations

Robust branch-cut-and-price for the Capacitated Minimum Spanning Tree problem over a large extended formulation

Robust Branch-Cut-and-Price Algorithms for Vehicle Routing Problems

Contact Info

Product

Resources

About