577 research outputs found
Exact and heuristic approaches for multi-component optimisation problems
Modern real world applications are commonly complex, consisting of multiple subsystems
that may interact with or depend on each other. Our case-study about wave
energy converters (WEC) for the renewable energy industry shows that in such a
multi-component system, optimising each individual component cannot yield global
optimality for the entire system, owing to the influence of their interactions or the
dependence on one another. Moreover, modelling a multi-component problem is
rarely easy due to the complexity of the issues, which leads to a desire for existent
models on which to base, and against which to test, calculations. Recently,
the travelling thief problem (TTP) has attracted significant attention in the Evolutionary
Computation community. It is intended to offer a better model for multicomponent
systems, where researchers can push forward their understanding of
the optimisation of such systems, especially for understanding of the interconnections
between the components. The TTP interconnects with two classic NP-hard
problems, namely the travelling salesman problem and the 0-1 knapsack problem,
via the transportation cost that non-linearly depends on the accumulated weight
of items. This non-linear setting introduces additional complexity. We study this
nonlinearity through a simplified version of the TTP - the packing while travelling
(PWT) problem, which aims to maximise the total reward for a given travelling tour.
Our theoretical and experimental investigations demonstrate that the difficulty of a
given problem instance is significantly influenced by adjusting a single parameter,
the renting rate, which prompted our method of creating relatively hard instances
using simple evolutionary algorithms. Our further investigations into the PWT
problem yield a dynamic programming (DP) approach that can solve the problem in
pseudo polynomial time and a corresponding approximation scheme. The experimental
investigations show that the new approaches outperform the state-of-the-art
ones. We furthermore propose three exact algorithms for the TTP, based on the DP
of the PWT problem. By employing the exact DP for the underlying PWT problem
as a subroutine, we create a novel indicator-based hybrid evolutionary approach for
a new bi-criteria formulation of the TTP. This hybrid design takes advantage of the
DP approach, along with a number of novel indicators and selection mechanisms
to achieve better solutions. The results of computational experiments show that the
approach is capable to outperform the state-of-the-art results.Thesis (Ph.D.) -- University of Adelaide, School of Computer Science, 201
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
Stochastic Combinatorial Optimization via Poisson Approximation
We study several stochastic combinatorial problems, including the expected
utility maximization problem, the stochastic knapsack problem and the
stochastic bin packing problem. A common technical challenge in these problems
is to optimize some function of the sum of a set of random variables. The
difficulty is mainly due to the fact that the probability distribution of the
sum is the convolution of a set of distributions, which is not an easy
objective function to work with. To tackle this difficulty, we introduce the
Poisson approximation technique. The technique is based on the Poisson
approximation theorem discovered by Le Cam, which enables us to approximate the
distribution of the sum of a set of random variables using a compound Poisson
distribution.
We first study the expected utility maximization problem introduced recently
[Li and Despande, FOCS11]. For monotone and Lipschitz utility functions, we
obtain an additive PTAS if there is a multidimensional PTAS for the
multi-objective version of the problem, strictly generalizing the previous
result.
For the stochastic bin packing problem (introduced in [Kleinberg, Rabani and
Tardos, STOC97]), we show there is a polynomial time algorithm which uses at
most the optimal number of bins, if we relax the size of each bin and the
overflow probability by eps.
For stochastic knapsack, we show a 1+eps-approximation using eps extra
capacity, even when the size and reward of each item may be correlated and
cancelations of items are allowed. This generalizes the previous work [Balghat,
Goel and Khanna, SODA11] for the case without correlation and cancelation. Our
algorithm is also simpler. We also present a factor 2+eps approximation
algorithm for stochastic knapsack with cancelations. the current known
approximation factor of 8 [Gupta, Krishnaswamy, Molinaro and Ravi, FOCS11].Comment: 42 pages, 1 figure, Preliminary version appears in the Proceeding of
the 45th ACM Symposium on the Theory of Computing (STOC13
Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization
International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM
A Branch-and-Price Algorithm for Bin Packing Problem
Bin Packing Problem examines the minimum number of identical bins needed to pack a set of items of various sizes. Employing branch-and-bound and column generation usually requires designation of the problem-specific branching rules compatible with the nature of the pricing sub-problem of column generation, or alternatively it requires determination of the k-best solutions of knapsack problem at level kth of the tree. Instead, we present a new approach to deal with the pricing sub-problem of column generation which handles two-dimensional knapsack problems. Furthermore, a set of new upper bounds for Bin Packing Problem is introduced in this work which employs solutions of the continuous relaxation of the set-covering formulation of Bin Packing Problem. These high quality upper bounds are computed inexpensively and dominate the ones generated by state-of-the-art methods
Dynamic Programming to Solve Picking Schedule at the Tea Plantation
The tea picking schedule at PT Perkebunan Ciater isset to be the same for all plantation blocks. Infact, the altitude from sea level and the pruning age of each plantation block is different, this results in thedifference of buds’growth. The implementation of the same pick-ing schedule causesthe quality and quantity of tea buds often couldnot be fulfilled. This research is to determine the precise picking schedule by considering the buds’growth of each plantation block. Two steps are implemented to solve the problem. The first step is to look for picking period and the pattern of buds’quality for each plantation block, which corresponds to the altitude of the location and the pruning age. The regression method is applied in this first step. The buds’quality pattern is then used to determine the cost of de-creasing buds’quality and the costs of the budsthat left in the plantation. The second step is to develop the picking schedule using dy-namic programming, which minimizesthe total cost of picking. In addition to this, we also develop a rolling schedule, which schedule time interval is three days. The modelresults show that the proposed schedule givesa better total cost than the current schedule and the buds’ quality target is easier to achieve.
Keywords:Dynamic Programming; Minimizes Cost; Picking Schedul
Models and algorithms for decomposition problems
This thesis deals with the decomposition both as a solution method and as a problem itself. A decomposition approach can be very effective for mathematical problems presenting a specific structure in which the associated matrix of coefficients is sparse and it is diagonalizable in blocks. But, this kind of structure may not be evident from the most natural formulation of the problem. Thus, its coefficient matrix may be preprocessed by solving a structure detection problem in order to understand if a decomposition method can successfully be applied. So, this thesis deals with the k-Vertex Cut problem, that is the problem of finding the minimum subset of nodes whose removal disconnects a graph into at least k components, and it models relevant applications in matrix decomposition for solving systems of equations by parallel computing. The capacitated k-Vertex Separator problem, instead, asks to find a subset of vertices of minimum cardinality the deletion of which disconnects a given graph in at most k shores and the size of each shore must not be larger than a given capacity value. Also this problem is of great importance for matrix decomposition algorithms.
This thesis also addresses the Chance-Constrained Mathematical Program that represents a significant example in which decomposition techniques can be successfully applied. This is a class of stochastic optimization problems in which the feasible region depends on the realization of a random variable and the solution must optimize a given objective function while belonging to the feasible region with a probability that must be above a given value. In this thesis, a decomposition approach for this problem is introduced.
The thesis also addresses the Fractional Knapsack Problem with Penalties, a variant of the knapsack problem in which items can be split at the expense of a penalty depending on the fractional quantity
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