Jürgen Rietz

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.

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.

On the extremality of maximal dual feasible functions

Abstract Dual feasible functions have been used to compute bounds and valid inequalities for combinatorial optimization problems. Here, we analyze the properties of some of the best functions proposed so far. Additionally, we provide new results for composed functions. These results will allow improving the computation of bounds and valid inequalities.

On the Properties of General Dual-Feasible Functions

Dual-feasible functions have been used to compute fast lower bounds and valid inequalities for integer linear optimization problems. However, almost all the functions proposed in the literature are defined only for positive arguments, which restricts considerably their applicability. The characteristics and properties of dual-feasible functions with general domains remain mostly unknown. In this paper, we show that extending these functions to negative arguments raises many issues. We explore these functions in depth with a focus on maximal functions, i.e. the family of non-dominated functions. The knowledge of these properties is fundamental to derive good families of general maximal dual-feasible functions that might lead to strong cuts for integer linear optimization problems and strong lower bounds for combinatorial optimization problems with knapsack constraints.

Worst-case analysis of maximal dual feasible functions

Dual feasible functions have been used to compute fast lower bounds and valid inequalities for integer linear problems. In this paper, we analyze the worst-case performance of the lower bounds provided by some of the best functions proposed in the literature. We describe some worst-case examples for these functions, and we report on new results concerning the best parameter choice for one of these functions.

Constructing general dual-feasible functions

Dual-feasible functions have proved to be very effective for generating fast lower bounds and valid inequalities for integer linear programs with knapsack constraints. However, a significant limitation is that they are defined only for positive arguments. Extending the concept of dual-feasible function to the general domain and range R is not straightforward. In this paper, we propose the first construction principles to obtain general functions with domain and range R , and we show that they lead to non-dominated maximal functions.

An exact approach based on a new pseudo-polynomial network flow model for integrated planning and scheduling

The resolution of planning and scheduling problems in a coordinated way within the supply chain is very challenging. In this paper, we address the integration of medium-term production planning and short-term scheduling models. We particularly focus on a specific problem defined on parallel machines that has recently been explored in the literature. The problem is characterized by a set of jobs that can be processed only from a given release date onward, and which should be finished at a given due date. At a first stage, the problem consists in assigning the jobs to consecutive time periods within the planning horizon, while at a second stage, the jobs have to be scheduled on the available machines.Our contribution consists in the description and analysis of a new detailed scheduling model based on a pseudo-polynomial network flow formulation that can be used to exactly solve real size instances. We explore different strategies to simplify the model and reduce its number of constraints. To evaluate the performance of our approaches, we report an extensive set of computational experiments on benchmark instances from the literature. The results obtained show that our approach outperforms, on some classes of instances, other state-of-the-art methods described recently in the literature. HighlightsNew pseudo-polynomial network flow model for integrated planning and scheduling.We describe procedures to simplify the model.The problem arises in the area of production planning.Our model improves the resolution of the problem for different classes of instances.

Linear and Integer Programming

Integer Programming (IP) is a modelling tool that has been widely applied in the last decades to obtain solutions for complex real problems, as those that arise in cutting and packing, location, routing and many other areas.

Other Applications in General Integer Programming

In this chapter, we briefly review an alternative application of dual-feasible functions in general integer programming. We explore these functions in particular to derive valid inequalities for integer programs. Since the notion of superadditivity is essential for this purpose, we start by reviewing superadditivity in the scope of valid inequalities. Different examples are provided with alternative families of dualfeasible functions. We discuss also the difference between the valid inequalities derived by dual-feasible functions and the well-known Chvatal-Gomory cuts.

Applications for Cutting and Packing Problems

Dual-feasible functions have been designed specifically for the cutting-stock problem. As shown in Chap. 1, they arise naturally from the dual of the classical formulation of Gilmore and Gomory for this problem. Since many problems can be modeled using a similar formulation, it makes sense to explore the concept of dual-feasible function within a more general class of applications. A first approach is to considermulti-dimensional dual-feasible functions,which can be used to derive lower bounds for the vector packing problem. Here, we also consider different packing problems with more complicated subproblems such as multi-dimensional orthogonal packing and packing with conflicts. Dual-feasible functions can still be derived in these cases.