Discrete Optimization Methods for Packing Problems in Two and Three Dimensions — With Applications in the Textile and Car Manufacturing Industries

Abstract

This talk surveys two- and three-dimensional packing problems in the textile and leather manufacturing industry as well as in the automobile industry and presents methods for solving such problems. We solve these problems with methods from combinatorial optimization. Our research leads to software that is used in the respective industrial branches.

On leather, the problem is to place stencils (for objects such as sofas, car seats, shoes etc.) on a leather hide such as to minimize the waste and to obey certain restrictions pertaining to leather quality. Since runtime requirements are tough, we are using a greedy placement strategy here that employs a shape fitting method based on a variant of geometric hashing.Work on leather is joint with Jörg Heistermann, Ralf Heckmann, and Birgit Fromme.

On textiles, the problem is analogous to the leather case. Here, the fabric is rectangular, rotations of the stencils are restricted, and quality restrictions are replaced with a wide variety of constraints pertaining to patterns on the fabric. Cutting images can be reused in this case, and therefore more runtime is allowed. The most successful algorithmic variant for this case employs a fast randomized strategy for generating cutting images. This strategy is based on Minkowski sums. We can generate up to 1 Mio images per hour, the best of which is postoptimized with simulated annealing or a linear-programming method and then offered as the solution. On fabric, we can also compute lower bounds. For this purpose, we extend a branch-and-bound technique by Milenkovic to compute bounds that are quite tight (below 1 percentage point of waste) for pants and less tight (within several percentage points) on jackets. We also developed an algorithm that uses a database of cutting images to generate cutting images based on their similarity to already computed high-quality cutting images. By continually extending the database with cutting images computed by the algorithm, the algorithm is able to learn. Work on textiles is joint with Ralf Heckmann at GMD-SCAI. Research on textiles has been partially supported by the German National Science Foundation (DFG).

Arrangement problems in the car manufacturing industry take many forms and are generally not yet very accessible to automation. The reason is the multitude of constraints on these problems (geometric, electromagnetic, thermal, maintenance, ergonomic) and the elusiveness of the cost function. We have chosen an approach in stages to make these problem more amenable to automation. In stage 1, we extended our approach to chip layout to three dimensions, in order to pack boxes with electronic modules in cars. Here the shapes are brick-like. In stage 2, we develop methods based on combinatorial optimization that pack more general - even nonconvex - 3D shapes. We use a two-phase method here. In the first phase we generate variants of the relative position and orientation of these shapes. In the second stage we apply a local compaction method that is able to change the orientation of the objects within limits. Here, we extend a method by Milenkovic to three dimensions. In the third stage, we integrate the methods that have been developed and propose scenarios for their application in the automobile industry. Packing electronic parts, packing parts in trunks of cars, and inserting modules like dynamos etc. into preplaced engine spaces are examples of such scenarios. Work on automobiles is in progress and is joint with Mike Schäfer at the Department of Computer Science, University of Bonn. Research on automobiles has been partially supported by the German Federal Ministry for Education and Research (BMBF).