21 research outputs found
A branch-and-price algorithm for the temporal bin packing problem
We study an extension of the classical Bin Packing Problem, where each item consumes the bin capacity during a given time window that depends on the item itself. The problem asks for finding the minimum number of bins to pack all the items while respecting the bin capacity at any time instant. A polynomial-size formulation, an exponential-size formulation, and a number of lower and upper bounds are studied. A branch-and-price algorithm for solving the exponential-size formulation is introduced. An overall algorithm combining the different methods is then proposed and tested through extensive computational experiments
Arc flow formulations based on dynamic programming: Theoretical foundations and applications
Network flow formulations are among the most successful tools to solve optimization problems. Such formulations correspond to determining an optimal flow in a network. One particular class of network flow formulations is the arc flow, where variables represent flows on individual arcs of the network. For NP-hard problems, polynomial-sized arc flow models typically provide weak linear relaxations and may have too much symmetry to be efficient in practice. Instead, arc flow models with a pseudo-polynomial size usually provide strong relaxations and are efficient in practice. The interest in pseudo-polynomial arc flow formulations has grown considerably in the last twenty years, in which they have been used to solve many open instances of hard problems. A remarkable advantage of pseudo-polynomial arc flow models is the possibility to solve practical-sized instances directly by a Mixed Integer Linear Programming solver, avoiding the implementation of complex methods based on column generation. In this survey, we present theoretical foundations of pseudo-polynomial arc flow formulations, by showing a relation between their network and Dynamic Programming (DP). This relation allows a better understanding of the strength of these formulations, through a link with models obtained by Dantzig-Wolfe decomposition. The relation with DP also allows a new perspective to relate state-space relaxation methods for DP with arc flow models. We also present a dual point of view to contrast the linear relaxation of arc flow models with that of models based on paths and cycles. To conclude, we review the main solution methods and applications of arc flow models based on DP in several domains such as cutting, packing, scheduling, and routing
Enhanced pseudo-polynomial formulations for bin packing and cutting stock problems
We study pseudopolynomial formulations for the classical bin packing and cutting stock problems. We first propose an overview of dominance and equivalence relations among the main pattern-based and pseudopolynomial formulations from the literature. We then introduce reflect, a new formulation that uses just half of the bin capacity to model an instance and needs significantly fewer constraints and variables than the classical models. We propose upper- and lower-bounding techniques that make use of column generation and dual information to compensate reflect weaknesses when bin capacity is too high. We also present nontrivial adaptations of our techniques that solve two interesting problem variants, namely the variable-sized bin packing problem and the bin packing problem with item fragmentation. Extensive computational tests on benchmark instances show that our algorithms achieve state of the art results on all problems, improving on previous algorithms and finding several new proven optimal solutions
Solving Bin Packing Problems Using VRPSolver Models
International audienceWe propose branch-cut-and-price algorithms for the classic bin packing problem and also for the following related problems: vector packing, variable sized bin packing and variable sized bin packing with optional items. The algorithms are defined as models for VRPSolver, a generic solver for vehicle routing problems. In that way, a simple parameterization enables the use of several branch-cut-and-price advanced elements: automatic stabilization by smoothing, limited-memory rank-1 cuts, enumeration, hierarchical strong branching and limited discrepancy search diving heuristics. As an original theoretical contribution, we prove that the branching over accumulated resource consumption (GĂ©linas et al. 1995), that does not increase the difficulty of the pricing subproblem, is sufficient for those bin packing models. Extensive computational results on instances from the literature show that the VRPSolver models have a performance that is very robust over all those problems, being often superior to the existing exact algorithms on the hardest instances. Several instances could be solved to optimality for the first time
A branch, price, and cut approach to solving the maximum weighted independent set problem
The maximum weight-independent set problem (MWISP) is one of the most
well-known and well-studied NP-hard problems in the field of combinatorial
optimization.
In the first part of the dissertation, I explore efficient branch-and-price (B&P)
approaches to solve MWISP exactly. B&P is a useful integer-programming tool for
solving NP-hard optimization problems. Specifically, I look at vertex- and edge-disjoint
decompositions of the underlying graph. MWISPĂÂąĂĂs on the resulting subgraphs are less
challenging, on average, to solve. I use the B&P framework to solve MWISP on the
original graph G using these specially constructed subproblems to generate columns. I
demonstrate that vertex-disjoint partitioning scheme gives an effective approach for
relatively sparse graphs. I also show that the edge-disjoint approach is less effective than
the vertex-disjoint scheme because the associated DWD reformulation of the latter
entails a slow rate of convergence.
In the second part of the dissertation, I address convergence properties associated
with Dantzig-Wolfe Decomposition (DWD). I discuss prevalent methods for improving the rate of convergence of DWD. I also implement specific methods in application to the
edge-disjoint B&P scheme and show that these methods improve the rate of
convergence.
In the third part of the dissertation, I focus on identifying new cut-generation
methods within the B&P framework. Such methods have not been explored in the
literature. I present two new methodologies for generating generic cutting planes within
the B&P framework. These techniques are not limited to MWISP and can be used in
general applications of B&P. The first methodology generates cuts by identifying faces
(facets) of subproblem polytopes and lifting associated inequalities; the second
methodology computes Lift-and-Project (L&P) cuts within B&P. I successfully
demonstrate the feasibility of both approaches and present preliminary computational
tests of each