137,126 research outputs found
Using Branch-and-Price to Find High Quality Solutions Quickly
We develop an exact solution approach for integer programs that produces high-
quality solutions quickly by solving well-chosen restrictions of the problem. Column generation is used both for generating these problem restrictions and for producing bounds on the value of an optimal solution to the problem. Obtaining primal solutions by solving problem restrictions also provides an easy way to search for improved solutions in the neighborhood of the current best solution. The overall approach is parallelized and computational experiments demonstrate its efficacy. An application to inventory routing is presented
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
Industrial and Tramp Ship Routing Problems: Closing the Gap for Real-Scale Instances
Recent studies in maritime logistics have introduced a general ship routing
problem and a benchmark suite based on real shipping segments, considering
pickups and deliveries, cargo selection, ship-dependent starting locations,
travel times and costs, time windows, and incompatibility constraints, among
other features. Together, these characteristics pose considerable challenges
for exact and heuristic methods, and some cases with as few as 18 cargoes
remain unsolved. To face this challenge, we propose an exact branch-and-price
(B&P) algorithm and a hybrid metaheuristic. Our exact method generates
elementary routes, but exploits decremental state-space relaxation to speed up
column generation, heuristic strong branching, as well as advanced
preprocessing and route enumeration techniques. Our metaheuristic is a
sophisticated extension of the unified hybrid genetic search. It exploits a
set-partitioning phase and uses problem-tailored variation operators to
efficiently handle all the problem characteristics. As shown in our
experimental analyses, the B&P optimally solves 239/240 existing instances
within one hour. Scalability experiments on even larger problems demonstrate
that it can optimally solve problems with around 60 ships and 200 cargoes
(i.e., 400 pickup and delivery services) and find optimality gaps below 1.04%
on the largest cases with up to 260 cargoes. The hybrid metaheuristic
outperforms all previous heuristics and produces near-optimal solutions within
minutes. These results are noteworthy, since these instances are comparable in
size with the largest problems routinely solved by shipping companies
Minimizing the average distance to a closest leaf in a phylogenetic tree
When performing an analysis on a collection of molecular sequences, it can be
convenient to reduce the number of sequences under consideration while
maintaining some characteristic of a larger collection of sequences. For
example, one may wish to select a subset of high-quality sequences that
represent the diversity of a larger collection of sequences. One may also wish
to specialize a large database of characterized "reference sequences" to a
smaller subset that is as close as possible on average to a collection of
"query sequences" of interest. Such a representative subset can be useful
whenever one wishes to find a set of reference sequences that is appropriate to
use for comparative analysis of environmentally-derived sequences, such as for
selecting "reference tree" sequences for phylogenetic placement of metagenomic
reads. In this paper we formalize these problems in terms of the minimization
of the Average Distance to the Closest Leaf (ADCL) and investigate algorithms
to perform the relevant minimization. We show that the greedy algorithm is not
effective, show that a variant of the Partitioning Among Medoids (PAM)
heuristic gets stuck in local minima, and develop an exact dynamic programming
approach. Using this exact program we note that the performance of PAM appears
to be good for simulated trees, and is faster than the exact algorithm for
small trees. On the other hand, the exact program gives solutions for all
numbers of leaves less than or equal to the given desired number of leaves,
while PAM only gives a solution for the pre-specified number of leaves. Via
application to real data, we show that the ADCL criterion chooses chimeric
sequences less often than random subsets, while the maximization of
phylogenetic diversity chooses them more often than random. These algorithms
have been implemented in publicly available software.Comment: Please contact us with any comments or questions
The Vehicle Routing Problem with Service Level Constraints
We consider a vehicle routing problem which seeks to minimize cost subject to
service level constraints on several groups of deliveries. This problem
captures some essential challenges faced by a logistics provider which operates
transportation services for a limited number of partners and should respect
contractual obligations on service levels. The problem also generalizes several
important classes of vehicle routing problems with profits. To solve it, we
propose a compact mathematical formulation, a branch-and-price algorithm, and a
hybrid genetic algorithm with population management, which relies on
problem-tailored solution representation, crossover and local search operators,
as well as an adaptive penalization mechanism establishing a good balance
between service levels and costs. Our computational experiments show that the
proposed heuristic returns very high-quality solutions for this difficult
problem, matches all optimal solutions found for small and medium-scale
benchmark instances, and improves upon existing algorithms for two important
special cases: the vehicle routing problem with private fleet and common
carrier, and the capacitated profitable tour problem. The branch-and-price
algorithm also produces new optimal solutions for all three problems
An efficient memetic, permutation-based evolutionary algorithm for real-world train timetabling
Train timetabling is a difficult and very tightly constrained combinatorial
problem that deals with the construction of train schedules. We focus on the
particular problem of local reconstruction of the schedule following a small
perturbation, seeking minimisation of the total accumulated delay by adapting
times of departure and arrival for each train and allocation of resources
(tracks, routing nodes, etc.). We describe a permutation-based evolutionary
algorithm that relies on a semi-greedy heuristic to gradually reconstruct the
schedule by inserting trains one after the other following the permutation.
This algorithm can be hybridised with ILOG commercial MIP programming tool
CPLEX in a coarse-grained manner: the evolutionary part is used to quickly
obtain a good but suboptimal solution and this intermediate solution is refined
using CPLEX. Experimental results are presented on a large real-world case
involving more than one million variables and 2 million constraints. Results
are surprisingly good as the evolutionary algorithm, alone or hybridised,
produces excellent solutions much faster than CPLEX alone
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