Heuristics for Multidimensional Packing Problems

In this thesis we consider solution methods for packing problems. Packing problems occur in manydifferent situations both directly in the industry and as sub-problems of other problems. High-qualitysolutions for problems in the industrial sector may be able to reduce transportation and productioncosts significantly. For packing problems in general are given a set of items and one of more contain-ers. The items must be placed within the container such that some objective is optimized and the itemsdo not overlap. Items and container may be rectangular or irregular (e.g. polygons and polyhedra)and may be defined in any number of dimensions. Solution methods are based on theory from bothcomputational geometry and operations research.The scientific contributions of this thesis are presented in the form of six papers and a sectionwhich introduces the many problem types and recent solution methods. Two important problem vari-ants are the knapsack packing problem and the strip-packing problem. In the knapsack packing prob-lem, each item is given a profit value, and the problem asks for the subset with maximal profit thatcan be placed within one container. The strip-packing problem asks for a minimum height containerrequired for the items. The main contributions of the thesis are three new heuristics for strip-packingand knapsack packing problems where items are both rectangular and irregular.In the two first papers we describe a heuristic for the multidimensional strip-packing problem thatis based on a relaxed placement principle. The heuristic starts with a random overlapping placement ofitems and large container dimensions. From the overlapping placement overlap is reduced iterativelyuntil a non-overlapping placement is found and a new problem is solved with a smaller container size.This is repeated until a time-limit is reached, and the smallest container for which a non-overlappingplacement was found is returned as solution. In each iteration, a single item is translated parallel toone of the coordinate axes to the position that reduces the overlap the most. Experimental resultsof this heuristic are among the best published in the literature both for twoand three-dimensionalstrip-packing problems for irregular shapes.In the third paper, we introduce a heuristic for twoand three-dimensional rectangular knapsackpacking problems. The two-dimensional heuristic uses the sequence pair representation and a novelrepresentation called sequence triple is introduced for the three-dimensional variant. Experiments forthe two-dimensional knapsack packing problem are on-par with the best published in the literatureand experiments for the three-dimensional variant are promising.A heuristic for a three-dimensional knapsack packing problem involving furniture is presented inthe fourth paper. The heuristic is based on a variety of techniques including tree-search, wall-building,and sequential placement. The solution process includes considerations regarding stability and loadbearing strength of items. The heuristic was developed in collaboration with an industrial partner andis now being used to solve hundreds of problems every day as part of their planning process.A simple heuristic for optimizing a placement of items with respect to balance and moment of iner-tia is presented in the fifth paper. Ensuring that a loaded consignment of items are balanced throughouta container can reduce fuel consumption and prolong the life-span of vehicles. The heuristic can beused as a post-processing tool to reorganize an existing solution to a packing problem.A method for optimizing the placement of cylinders with spherical ends is presented in the lastpaper. The method can consider proximity constraints which can be used to describe how cylindersshould be placed relative to each other. The method is applied to problems where a placement ofcapsules must be found within a minimal spherical or box-shaped container and to problems wherea placement within a given arbitrarily container must be found. The method has applications forprediction of RNA tertiary structure.