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

The ideal trefoil knot

The most tight conformation of the trefoil knot found by the SONO algorithm is presented. Structure of the set of its self-contact points is analyzed.Comment: 11 pages, 8 figure

A Monte-Carlo approach to tool selection for sheet metal punching and nibbling

Selecting the best set of tools to produce certain geometrical shapes/features in sheet metal punching is one of the problems that has a great effect on product development time, cost and achieved quality. The trend nowadays is, where at all possible, to limit design to the use of standard tools. Such an option makes the problem of selecting the appropriate set of tools even more complex, especially when considering that sheet metal features can have a wide range of complex shapes. Another dimension of complexity is limited tool rack capacity. Thus, an inappropriate tool selection strategy will lead to punching inefficiency and may require frequent stopping of the machine and replacing the required tools, which is a rather expensive and time consuming exercise. This work demonstrates that the problem of selecting the best set of tools is actually a process of searching an explosive decision tree. The difficulty in searching such types of decision trees is that intermediate decisions do not necessarily reflect the total cost implication of carrying out such a decision. A new approach to solve such a complex optimisation problem using the Monte Carlo Simulation Methods has been introduced in this thesis. The aim of the present work was to establish the use of Monte Carlo methods as an "assumptions or rule free" baseline or benchmark for the assessment of search strategies. A number of case studies are given, where the feasibility of Monte Carlo Simulation Methods as an efficient and viable method to optimise such a complex optimisation problem is demonstrated. The use of a Monte Carlo approach for selecting the best set of punching tools, showed an interesting point, that is, the effect of dominant "one-to-one" feature/tool matches on the efficiency of the search. This naturally led on to the need of a search methodology that will be more efficient than the application of the Monte Carlo method alone. This thesis presents some interesting speculations for a hybrid approach to tool selection to achieve a better solution than the use of the Monte Carlo method alone to achieve the optimum solution in a shorter time