Cláudio Alves

A survey of dual-feasible and superadditive functions

Dual-feasible functions are valuable tools that can be used to compute both lower bounds for different combinatorial problems and valid inequalities for integer programs. Several families of functions have been used in the literature. Some of them were defined explicitly, and others not. One of the main objectives of this paper is to survey these functions, and to state results concerning their quality. We clearly identify dominant subsets of functions, i.e. those which may lead to better bounds or stronger cuts. We also describe different frameworks that can be used to create dual-feasible functions. With these frameworks, one can get a dominant function based on other ones. Two new families of dual-feasible functions obtained by applying these methods are proposed in this paper.We also performed a computational comparison on the relative strength of the functions presented in this paper for deriving lower bounds for the bin-packing problem and valid cutting planes for the pattern minimization problem. Extensive experiments on instances generated using methods described in the literature are reported. In many cases, the lower bounds are improved, and the linear relaxations are strengthened.

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.

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.

Multidimensional dual-feasible functions and fast lower bounds for the vector packing problem

In this paper, we address the 2-dimensional vector packing problem where an optimal layout for a set of items with two independent dimensions has to be found within the boundaries of a rectangle. Many practical applications in areas such as the telecommunications, transportation and production planning lead to this combinatorial problem. Here, we focus on the computation of fast lower bounds using original approaches based on the concept of dual-feasible functions.

An Exact Algorithm for Bilevel 0-1 Knapsack Problems

We propose a new exact method for solving bilevel 0-1 knapsack problems. A bilevel problem models a hierarchical decision process that involves two decision makers called the leader and the follower. In these processes, the leader takes his decision by considering explicitly the reaction of the follower. From an optimization standpoint, these are problems in which a subset of the variables must be the optimal solution of another (parametric) optimization problem. These problems have various applications in the field of transportation and revenue management, for example. Our approach relies on different components. We describe a polynomial time procedure to solve the linear relaxation of the bilevel 0-1 knapsack problem. Using the information provided by the solutions generated by this procedure, we compute a feasible solution (and hence a lower bound) for the problem. This bound is used together with an upper bound to reduce the size of the original problem. The optimal integer solution of the original problem is computed using dynamic programming. We report on computational experiments which are compared with the results achieved with other state-of-the-art approaches. The results attest the performance of our approach.

Theoretical investigations on maximal dual feasible functions

Dual feasible functions are used to get valid inequalities and lower bounds for integer linear problems. In this paper, we provide a simpler proof for maximality, and we describe new results concerning the extremality of functions from the literature. Extremal functions are a dominant class of dual feasible functions.

A branch-and-price-and-cut algorithm for the pattern minimization problem

In cutting stock problems, after an optimal (minimal stock usage) cutting plan has been devised, one might want to further reduce the operational costs by minimizing the number of setups. A setup operation occurs each time a different cutting pattern begins to be produced. The related optimization problem is known as the Pattern Minimization Problem, and it is particularly hard to solve exactly. In this paper, we present different techniques to strengthen a formulation proposed in the literature. Dual feasible functions are used for the first time to derive valid inequalities from different constraints of the model, and from linear combinations of constraints. A new arc flow formulation is also proposed. This formulation is used to define the branching scheme of our branch-and-price-and-cut algorithm, and it allows the generation of even stronger cuts by combining the branching constraints with other constraints of the model. The computational experiments conducted on instances from the literature show that our algorithm finds optimal integer solutions faster than other approaches. A set of computational results on random instances is also reported.

A hybrid heuristic for the multiple choice multidimensional knapsack problem

In this article, a new solution approach for the multiple choice multidimensional knapsack problem is described. The problem is a variant of the multidimensional knapsack problem where items are divided into classes, and exactly one item per class has to be chosen. Both problems are NP-hard. However, the multiple choice multidimensional knapsack problem appears to be more difficult to solve in part because of its choice constraints. Many real applications lead to very large scale multiple choice multidimensional knapsack problems that can hardly be addressed using exact algorithms. A new hybrid heuristic is proposed that embeds several new procedures for this problem. The approach is based on the resolution of linear programming relaxations of the problem and reduced problems that are obtained by fixing some variables of the problem. The solutions of these problems are used to update the global lower and upper bounds for the optimal solution value. A new strategy for defining the reduced problems is explored, together with a new family of cuts and a reformulation procedure that is used at each iteration to improve the performance of the heuristic. An extensive set of computational experiments is reported for benchmark instances from the literature and for a large set of hard instances generated randomly. The results show that the approach outperforms other state-of-the-art methods described so far, providing the best known solution for a significant number of benchmark instances.

New Stabilization Procedures for the Cutting Stock Problem

In this paper, we deal with a column generation-based algorithm for the classical cutting stock problem. This algorithm is known to have convergence issues, which are addressed in this paper. Our methods are based on the fact that there are interesting characterizations of the structure of the dual problem, and that a large number of dual solutions are known. First, we describe methods based on the concept of dual cuts, proposed by Valerio de Carvalho [Valerio de Carvalho, J. M. 2005. Using extra dual cuts to accelerate column generation. INFORMS J. Comput.17(2) 175--182]. We introduce a general framework for deriving cuts, and we describe a new type of dual cut that excludes solutions that are linear combinations of some other known solutions. We also explore new lower and upper bounds for the dual variables. Then we show how the prior knowledge of a good dual solution helps improve the results. It tightens the bounds around the dual values and makes the search converge faster if a solution is sought in its neighborhood first. A set of computational experiments on very hard instances is reported at the end of the paper; the results confirm the effectiveness of the methods proposed.