J. M. Valério de Carvalho

Exact solution of bin‐packing problems using column generation and branch‐and‐bound

We explore an arc flow formulation with side constraints for the one‐dimensionalbin‐packing problem. The model has a set of flow conservation constraints and a set ofconstraints that force the appropriate number of items to be included in the packing. Themodel is tightened by fixing some variables at zero level, to reduce the symmetry of thesolution space, and by introducing valid inequalities. The model is solved exactly using abranch‐and‐price procedure that combines deferred variable generation and branch‐and‐bound.At each iteration, the subproblem generates a set of columns, which altogether correspondto an attractive valid packing for a single bin. We describe this subproblem, and theway it is modified in the branch‐and‐bound phase, after the branching constraints are addedto the model. We report the computational times obtained in the solution of the bin‐packingproblems from the OR‐Library test data sets. The linear relaxation of this model provides astrong lower bound for the bin‐packing problem and leads to tractable branch‐and‐boundtrees for the instances under consideration.We explore an arc flow formulation with side constraints for the one‐dimensionalbin‐packing problem. The model has a set of flow conservation constraints and a set ofconstraints that force the appropriate number of items to be included in the packing. Themodel is tightened by fixing some variables at zero level, to reduce the symmetry of thesolution space, and by introducing valid inequalities. The model is solved exactly using abranch‐and‐price procedure that combines deferred variable generation and branch‐and‐bound.At each iteration, the subproblem generates a set of columns, which altogether correspondto an attractive valid packing for a single bin. We describe this subproblem, and theway it is modified in the branch‐and‐bound phase, after the branching constraints are addedto the model. We report the computational times obtained in the solution of the bin‐packingproblems from the OR‐Library test data sets. The linear relaxation of this model provides astrong lower bound for the bin‐packing problem and leads to tractable branch‐and‐boundtrees for the instances under consideration. Copyright Kluwer Academic Publishers 1999

LP models for bin packing and cutting stock problems

Abstract We review several linear programming (LP) formulations for the one-dimensional cutting stock and bin packing problems, namely, the models of Kantorovich, Gilmore–Gomory, onecut models, as in the Dyckhoff–Stadtler approach, position-indexed models, and a model derived from the vehicle routing literature. We analyse some relations between the corresponding LP relaxations, and their relative strengths, and refer how to derive branching schemes that can be used in the exact solution of these problems, using branch-and-price.

A branch-and-price algorithm for scheduling parallel machines with sequence dependent setup times

Abstract We consider the problem of scheduling n independent jobs on m unrelated parallel machines with sequence-dependent setup times and availability dates for the machines and release dates for the jobs to minimize a regular additive cost function. In this work, we develop a new branch-and-price optimization algorithm for the solution of this general class of parallel machines scheduling problems. A new column generation accelerating method, termed “ primal box ”, and a specific branching variable selection rule that significantly reduces the number of explored nodes are proposed. The computational results show that the approach solves problems of large size to optimality within reasonable computational time.

Solving the vehicle routing problem with time windows and multiple routes exactly using a pseudo-polynomial model

In this paper, we address a variant of the vehicle routing problem called the vehicle routing problem with time windows and multiple routes. It considers that a given vehicle can be assigned to more than one route per planning period. We propose a new exact algorithm for this problem. Our algorithm is iterative and it relies on a pseudo-polynomial network flow model whose nodes represent time instants, and whose arcs represent feasible vehicle routes. This algorithm was tested on a set of benchmark instances from the literature. The computational results show that our method is able to solve more instances than the only other exact method described so far in the literature, and it clearly outperforms this method in terms of computing time.

A stabilized branch-and-price-and-cut algorithm for the multiple length cutting stock problem

Many heuristic approaches have been proposed in the literature for the multiple length cutting stock problem, while only few results have been reported for exact solution algorithms. In this paper, we present a new branch-and-price-and-cut algorithm which outperforms other exact approaches proposed so far. Our conclusions are supported on many computational experiments conducted on instances from the literature. In the second part of the paper, we investigate the use of valid dual inequalities within the previous algorithm. We show how column generation can be accelerated in all the nodes of our branching tree using such inequalities. Until now, dual inequalities have been applied only for original linear programming relaxations. Their validity within a branch-and-bound procedure has never been investigated. Our computational experiments show that extending these inequalities to the whole branch-and-bound tree can significantly improve the convergence of our branch-and-price-and-cut algorithm.

Arc-flow model for the two-dimensional guillotine cutting stock problem

We describe an exact model for the two-dimensional cutting stock problem with two stages and the guillotine constraint. It is an integer linear programming (ILP) arc-flow model, formulated as a minimum flow problem, which is an extension of a model proposed by Valerio de Carvalho for the one dimensional case. In this paper, we explore the behavior of this model when it is solved with a commercial software, explicitly considering all its variables and constraints. We also derive a new family of cutting planes and a new lower bound, and consider some variants of the original problem. The model was tested on a set of real instances from the wood industry, with very good results. Furthermore the lower bounds provided by the linear programming relaxation of the model compare favorably with the lower bounds provided by models based on assignment variables.

An LP-based approach to a two-stage cutting stock problem

This paper describes a linear programming model for a two-stage cutting stock problem that arises in a make-to-order steel company. For technological reasons, there are two different cutting operations. Between the cutting operations, the intermediate rolls are processed to obtain the required characteristics. The minimization of the trim loss and the reduction of the number of set-ups are primary concerns in cutting stock problems. In this case, it is also important to minimize the number of intermediate rolls, in order to reduce the processing work. These objectives may be conflicting and a balance may be necessary. We present a simplified LP model, which is solved by a column generation technique. The subproblems are relatively large knapsack problems. Computational results for a microcomputer are presented.

Accelerating column generation for variable sized bin-packing problems

Abstract In this paper, we study different strategies to stabilize and accelerate the column generation method, when it is applied specifically to the variable sized bin-packing problem, or to its cutting stock counterpart, the multiple length cutting stock problem. Many of the algorithms for these problems discussed in the literature rely on column generation, processes that are known to converge slowly due to primal degeneracy and the excessive oscillations of the dual variables. In the sequel, we introduce new dual-optimal inequalities, and explore the principle of model aggregation as an alternative way of controlling the progress of the dual variables. Two algorithms based on aggregation are proposed. The first one relies on a row aggregated LP, while the second one solves iteratively sequences of doubly aggregated models. Working with these approximations, in the various stages of an iterative solution process, has proven to be an effective way of achieving faster convergence. The computational experiments were conducted on a broad range of instances, many of them published in the literature. They show a significant reduction of the number of column generation iterations and computing time.

A Note on Branch-and-Price Algorithms for the One-Dimensional Cutting Stock Problems

One algorithm proposed in “Branch-and-price algorithms for the one-dimensional cutting stock problems” (COAP, vol. 9, pp. 211–228, 1998) may not find an optimal solution to the cutting stock problem, as it is stated.One algorithm proposed in “Branch-and-price algorithms for the one-dimensional cutting stock problems” (COAP, vol. 9, pp. 211–228, 1998) may not find an optimal solution to the cutting stock problem, as it is stated.

A Computer Based Interactive Approach To A Two-Stage Cutting Stock Problem

AbstractIn this article, we present a computer interactive system, developed for a Portuguese steel manufacturer operating at the retail level. The production process includes not only cutting operations, which are performed in two different stages, but also intermediate processing.Minimization of trim loss and reduction of set-up are primary objective in every cutting stock problem. In this system, there is also another important objective, which is the minimization of the number of intermediate rolls. Because these objectives may be conflicting, a balance between them may be necessary. Another feature of the system is that the weight of the raw material rolls may vary slightly.New models were built and integrated in an interactive computer system, developed for a microcomputer. In each planning period, the system is used to establish a couple of dozens of operational cutting plans, one for each different raw material. The results obtained are slightly better than the previously obtained by experienced s...