427 research outputs found
Bin Packing and Related Problems: General Arc-flow Formulation with Graph Compression
We present an exact method, based on an arc-flow formulation with side
constraints, for solving bin packing and cutting stock problems --- including
multi-constraint variants --- by simply representing all the patterns in a very
compact graph. Our method includes a graph compression algorithm that usually
reduces the size of the underlying graph substantially without weakening the
model. As opposed to our method, which provides strong models, conventional
models are usually highly symmetric and provide very weak lower bounds.
Our formulation is equivalent to Gilmore and Gomory's, thus providing a very
strong linear relaxation. However, instead of using column-generation in an
iterative process, the method constructs a graph, where paths from the source
to the target node represent every valid packing pattern.
The same method, without any problem-specific parameterization, was used to
solve a large variety of instances from several different cutting and packing
problems. In this paper, we deal with vector packing, graph coloring, bin
packing, cutting stock, cardinality constrained bin packing, cutting stock with
cutting knife limitation, cutting stock with binary patterns, bin packing with
conflicts, and cutting stock with binary patterns and forbidden pairs. We
report computational results obtained with many benchmark test data sets, all
of them showing a large advantage of this formulation with respect to the
traditional ones
Cyclic transfers in school timetabling
In this paper we propose a neighbourhood structure based on sequential/cyclic moves and a cyclic transfer algorithm for the high school timetabling problem. This method enables execution of complex moves for improving an existing solution, while dealing with the challenge of exploring the neighbourhood efficiently. An improvement graph is used in which certain negative cycles correspond to the neighbours; these cycles are explored using a recursive method. We address the problem of applying large neighbourhood structure methods on problems where the cost function is not exactly the sum of independent cost functions, as it is in the set partitioning problem. For computational experiments we use four real world data sets for high school timetabling in the Netherlands and England.We present results of the cyclic transfer algorithm with different settings on these data sets. The costs decrease by 8â28% if we use the cyclic transfers for local optimization compared to our initial solutions. The quality of the best initial solutions are comparable to the solutions found in practice by timetablers
Cyclic transfers in school timetabling
In this paper we propose a neighbourhood structure based\ud
on sequential/cyclic moves and a Cyclic Transfer algorithm for the high school timetabling problem. This method enables execution of complex moves for improving an existing solution, while dealing with the challenge of exploring the neighbourhood efficiently. An improvement graph is used in which certain negative cycles correspond to the neighbours; these cycles are explored using a recursive method. We address the problem of applying large neighbourhood structure methods on problems where the cost function is not exactly the sum of independent cost functions, as it is in the set partitioning problem. For computational experiments we use four real world datasets for high school timetabling in the Netherlands and England. We present results of the cyclic transfer algorithm with different settings on these datasets. The costs decrease by 8% to 28% if we use the cyclic transfers for local optimization compared to our initial solutions. The quality of the best initial solutions are comparable to the solutions found in practice by timetablers
On a Clique-Based Integer Programming Formulation of Vertex Colouring with Applications in Course Timetabling
Vertex colouring is a well-known problem in combinatorial optimisation, whose
alternative integer programming formulations have recently attracted
considerable attention. This paper briefly surveys seven known formulations of
vertex colouring and introduces a formulation of vertex colouring using a
suitable clique partition of the graph. This formulation is applicable in
timetabling applications, where such a clique partition of the conflict graph
is given implicitly. In contrast with some alternatives, the presented
formulation can also be easily extended to accommodate complex performance
indicators (``soft constraints'') imposed in a number of real-life course
timetabling applications. Its performance depends on the quality of the clique
partition, but encouraging empirical results for the Udine Course Timetabling
problem are reported
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