Cutting circles and polygons from area-minimizing rectangles

• Authors: Josef Kallrath
• Venue: Journal of Global Optimization
• Year: 2009

A set of circles, rectangles, and convex polygons are to be cutfrom rectangular design plates to be produced, or from a set ofstocked rectangles of known geometric dimensions. The objective isto minimize the area of the design rectangles. The design platesare subject to lower and upper bounds of their widths and lengths.The objects are free of any orientation restrictions. If all nestedobjects fit into one design or stocked plate the problem isformulated and solved as a nonconvex nonlinear programming problem.If the number of objects cannot be cut from a single plate,additional integer variables are needed to represent the allocationproblem leading to a nonconvex mixed integer nonlinear optimizationproblem. This is the first time that circles and arbitrary convexpolygons are treated simultaneously in this context. We presentexact mathematical programming solutions to both the design andallocation problem. For small number of objects to be cut wecompute globally optimal solutions. One key idea in the developedNLP and MINLP models is to use separating hyperplanes to ensurethat rectangles and polygons do not overlap with each other or withthe circles. Another important idea used when dealing with severalresource rectangles is to develop a model formulation whichconnects the binary variables only to the variables representingthe center of the circles or the vertices of the polytopes but notto the non-overlap or shape constraints. We support the solutionprocess by symmetry breaking constraints. In addition we computelower bounds, which are constructed by a relaxed model in whicheach polygon is replaced by the largest circle fitting into thatpolygon. We have successfully applied several solution techniquesto solve this problem among them the Branch&Reduce OptimizationNavigator (BARON) and the LindoGlobal solver called from GAMS, and,as described in Rebennack et al. [21], a column enumerationapproach in which the columns represent the assignments. Goodfeasible solutions are computed within seconds or minutes usuallyduring preprocessing. In most cases they turn out to be globallyoptimal. For up to 10 circles, we prove global optimality up to agap of the order of 10-8 in short time. Cases with amodest number of objects, for instance, 6 circles and 3 rectangles,are also solved in short time to global optimality. For testinstances involving non-rectangular polygons it is difficult toobtain small gaps. In such cases we are content to obtain gaps ofthe order of 10%.