Abstract
In this paper an application of constraint logic programming (CLP) to the resolution of nesting problems is presented. Nesting problems are a special case of the cutting and packing problems, in which the pieces generally have non-convex shapes. Because of their combinatorial optimization nature, nesting problems have traditionally been tackled by heuristics and in the recent past by meta-heuristics. When trying to formulate nesting problems as linear programming models, to achieve global optimal solutions, the difficulty of dealing with the disjunction of constraints arises. On the contrary, CLP deals easily with this type of relationships among constraints. A CLP implementation for the nesting problem is described for convex and non-convex shapes. The concept of nofit polygon is used to deal with the geometric constraints inherent to all cutting and packing problems. Computational results are presented. © 2003 Wiley Periodicals, Inc.

Abstract

Abstract
We address the short-term production planning and scheduling problem coming from the glass container industry. A furnace melts the glass that is distributed to a set of parallel molding machines. Both furnace and machine idleness are not allowed. The resulting multi-machine multi-item continuous setup lotsizing problem with a common resource has sequence-dependent setup times and costs. Production losses are penalized in the objective function since we deal with a capital intensive industry. We present two mixed integer programming formulations for this problem, which are reduced to a network flow type problem. The two formulations are improved by adding valid inequalities that lead to good lower bounds. We rely on a Lagrangian decomposition based heuristic for generating good feasible solutions. We report computational experiments for randomly generated instances and for real-life data on the aforementioned problem, as well as on a discrete lotsizing and scheduling version.

Abstract
Gupta and Magnusson [The capacitated lot-sizing and scheduling problem with sequence-dependent setup costs and setup times. Computers and Operations Research 2005;32(4):727-47] develop a model for the single machine capacitated lot-sizing and scheduling problem (CLSP) with sequence dependent setup times and setup costs, incorporating all the usual features of setup carryovers. In this note we show that this model does not avoid disconnected subtours. A new set of constraints is added to the model to provide an exact formulation for this problem.

Abstract
Inspired by a case study, this paper reports a successful application of VNS to the production planning and scheduling problem that arises in the glass container industry. This is a multi-facility production system, where each facility has a set of furnaces where the glass paste is produced in order to meet the demand, being afterwards distributed to a set of parallel molding machines. Since the neighborhoods used are not nested, they are not ordered by increasing sizes, but by means of a new empirical measure to assess the distance between any two solutions. Neighborhood sizes decrease significantly through-out the search thus suggesting the use of a scheme in which efficiency is placed. over effectiveness in a first step, and the opposite in a second step. We test this variant as well as other two with a real-world problem instance from our case study.

Abstract
In production planning in the glass container industry, machine-dependent setup times and costs are incurred for switch overs from one product to another. The resulting multi-item capacitated lot-sizing problem has sequence-dependent setup times and costs. We present two novel linear mixed-integer programming formulations for this problem, incorporating all the necessary features of setup carryovers. The compact formulation has polynomially many constraints, whereas the stronger formulation uses an exponential number of constraints that can be separated in polynomial time. We also present a five-step heuristic that is effective both in finding a feasible solution (even for tightly capacitated instances) and in producing good solutions to these problems. We report computational experiments.