The one-dimensional cutting stock problem and the two-dimensional two-staged constrained guillotine cutting (knapsack) problem are considered. They can be formulated by column generation. This model has a very tight continuous relaxation which provides a good bound in an LP-based solution approach. We combine a branching scheme and a cutting plane algorithm using Gomory mixed-integer and strengthened Chvatal-Gomory cuts.

Abstract. Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations

. A cutting plane approach combining Chvatal-Gomory cutting planes with column generation is generalized for the case of multiple stock lengths in the one-dimensional cutting stock problem. Appropriate modications of the column generation procedure and the rounding heuristic are proposed. A comparison with the branch-and-price method for the problem with one stock type and representative test results are reported. Keywords: cutting, cutting planes, column generation, heuristics, branch-and-bound 1

We consider the three-stage two-dimensional bin packing problem (2BP) which occurs in real-world applications such as glass, paper, or steel cutting. We present new integer linear programming formulations: Models for a restricted version and the original version of the problem are developed. Both involve polynomial numbers of variables and constraints only and e#ectively avoid symmetries. Those models are solved using CPLEX. Furthermore, a branch-and-price (B&P) algorithm is presented for a set covering formulation of the unrestricted problem. We consider stabilizing the column generation process of the B&P algorithm using dual-optimal inequalities. Fast column generation is performed by applying a hierarchy of four methods: (a) a fast greedy heuristic, (b) an evolutionary algorithm, (c) solving a restricted form of the pricing problem using CPLEX, and finally (d) solving the complete pricing problem using CPLEX. Computational experiments on standard benchmark instances document the benefits of the new approaches: The restricted version of the ILP model can be used for quickly obtaining nearly optimal solutions. The unrestricted version is computationally more expensive. Column generation provides a strong lower bound for 3-stage 2BP. The combination of all four pricing algorithms and column generation stabilization in the proposed B&P framework yields the best results in terms of the average objective value, the average run-time, and the number of instances solved to proven optimality.

Many numerical computations reported in the literature show only a small difference between the optimal value of the one-dimensional cutting stock problem (1CSP) and that of the corresponding linear programming relaxation. Moreover, theoretical investigations have proven that this difference is smaller than 2 for a wide range of subproblems of the general 1CSP. In this paper we give a branch&bound algorithm to compute optimal solutions for instances of the 1CSP. Numerical results are presented of about 900 randomly generated instances with up to 100 small pieces and all of them are optimally solved.

The one-dimensional cutting stock problem is investigated with respect to the difference between the optimal function value of the discrete problem and its continnuous relaxation. A tighter bound for this gap is presented, followed by some non-IRUP constructions. Finally, instances with gap 7/6 are constructed, the largest gap known so far.

In case of the one-dimensional cutting stock problem (CSP) one can observe for any instance a very small gap between the integer optimal value and the continuous relaxation bound. These observations have initiated a series of investigations. An instance possesses the integer round-up property (IRUP) if its gap is smaller than 1. It is well-known that there exist instances of the CSP having a gap greater than 1 but there is not known any instance with gap at least 2. In this paper we present families of non-IRUP instances and give methods to construct such instances. Furthermore, an equivalence relation for instances of the CSP is considered to become independent from the real size parameters.

The primary objective in cutting and packing problems is trim loss or material input minimization (in stock cutting) or value maximization (in knapsack-type problems). However, in real-life production we usually have many other objectives (costs) and constraints. Probably the most complex auxiliary criteria of a solution are the number of different cutting patterns (setups) and the maximum number of open stacks during the cutting process. There are

The polyhedral approach is one of the most powerful techniques available for solving hard combinatorial optimization problems. The main idea behind the technique is to consider the linear relaxation of the integer combinatorial optimization problem, and try to iteratively strengthen the linear formulation by adding violated strong valid inequalities, i.e., inequalities that are violated by the current fractional solution but satisfied by all feasible solutions, and that define high-dimensional faces, preferably facets, of the convex hull of feasible solutions. If we have the complete description of the convex hull of feasible solutions all extreme points of this formulation are integral, which means that we can solve the problem as a linear programming problem. Linear programming problems are known to be computationally easy. In Part I of this article we discuss theoretical aspects of polyhedral techniques. Here we will mainly concentrate on the computational aspects. In particular we ...

Let b ∈ Z d be an integer conic combination of a finite set of integer vectors X ⊂ Z d. In this note we provide upper bounds on the size of a smallest subset �X ⊆ X such that b is an integer conic combination of elements of �X. We apply our bounds to general integer programming and to the cutting stock problem and provide an NP certificate for the latter, whose existence has not been known so far.