Packing and Cutting Stone Blocks Based on the Nonlinear Programming of Tree Cases

Typically, dimension stones, commonly called stone blocks, are cut into multiple small cuboid stones so that multiple sculptures can be produced. To use the stone block as efficiently as possible, it is essential to pack these small cuboids in each stone block as efficiently as possible while satisfying the limitations of the machining. This paper describes methods for packing and cutting stone blocks using nonlinear programming that generate sets of trees, which are also called forests, that decide the packing layout of the small cuboids inside the block. The containers and elements have their own prices and values, respectively. The elements can be translated to the corners of the containers or to the corners of the elements that are already in the containers, if the elements are not outside the containers after the translation. Then, the problem can be interpreted as finding the best forest that packs the elements as efficiently as possible at the lowest total price of containers, which is a subset of all containers. The formula for the score that defines the compactness of the packing is in this paper. The user can define the number of forests so that parallel computing methods can be applied. Each forest is generated randomly. Two different packing methods are introduced: simple packing and slab packing. Simple packing is based on a non-guillotine cutting method and slab packing is a guillotine cutting method for realistic scenarios, such as scenarios with machining limitations. By using this method, it is possible to plan the cutting in a digital environment, which is not possible when using the traditional method with physical templates. Furthermore, by restricting the rotation of the elements, it is possible to make the elements follow the horizontal vein direction of the stone blocks, which is a common vein direction in travertine.