Solving Irregular Strip Packing Problems With Free Rotations Using Separation Lines

Solving nesting problems or irregular strip packing problems is to position polygons in a fixed width and unlimited length strip, obeying polygon integrity containment constraints and non-overlapping constraints, in order to minimize the used length of the strip. To ensure non-overlapping, we used separation lines. A straight line is a separation line if given two polygons, all vertices of one of the polygons are on one side of the line or on the line, and all vertices of the other polygon are on the other side of the line or on the line. Since we are considering free rotations of the polygons and separation lines, the mathematical model of the studied problem is nonlinear. Therefore, we use the nonlinear programming solver IPOPT (an algorithm of interior points type), which is part of COIN-OR. Computational tests were run using established benchmark instances and the results were compared with the ones obtained with other methodologies in the literature that use free rotation

FO(FD): Extending classical logic with rule-based fixpoint definitions

We introduce fixpoint definitions, a rule-based reformulation of fixpoint constructs. The logic FO(FD), an extension of classical logic with fixpoint definitions, is defined. We illustrate the relation between FO(FD) and FO(ID), which is developed as an integration of two knowledge representation paradigms. The satisfiability problem for FO(FD) is investigated by first reducing FO(FD) to difference logic and then using solvers for difference logic. These reductions are evaluated in the computation of models for FO(FD) theories representing fairness conditions and we provide potential applications of FO(FD).Comment: Presented at ICLP 2010. 16 pages, 1 figur

Mathematical Models for Minimizing Total Tardiness on Parallel Additive Manufacturing Machines

In this research we tackle the scheduling problem in additive manufacturing for unrelated parallel machines. Both the nesting and scheduling aspects are considered. Parts have several alternative build orientations. The goal is to minimize the total tardiness of parts. We propose a mixed-integer linear programming model which considers the nesting subproblem as a 2D bin-packing problem, as well as a model which simplifies the nesting subproblem to a 1D bin-packing problem. The computational efficiency and properties of the proposed models are investigated by numerical experiments. Results show that the total tardiness optimization significantly increases the complexity of the problem, only the simple instances are solved optimally, whereas the makespan variant is able to solve all testing instances. Using the 1D bin-packing simplification allows for solving more instances to optimality, but with a risk of obtaining nesting-infeasibility. We also observed the compromise between the total tardiness and makespan objectives, which originates from the dilemma of “packing more parts to benefit from the common machine setup/recoating time” or “packing less parts to maintain the flexibility for handling distributed duedates”