UNIVERSITY OF SOUTHAMPTON FACULTY OF BUSINESS, LAW AND ART Southampton Business School From archaeology to 3D printing: Packing problems in the three dimensions by Carlos Lamas Fern´andez Thesis for the degree of Doctor of Philosophy June 2018 UNIVERSITY OF SOUTHAMPTON ABSTRACT FACULTY OF BUSINESS, LAW AND ART Southampton Business School Doctor of Philosophy FROM ARCHAEOLOGY TO 3D PRINTING: PACKING PROBLEMS IN THE THREE DIMENSIONS by Carlos Lamas Fern´andez This thesis is a study on three cutting and packing problems involving irregular items. These problems are highly relevant in areas such as transportation, additive manufac- turing or the garment industry. We investigate a special type of one-dimensional bin packing problem appearing in the industry; a novel problem in two dimensions entailing irregular shapes and free rotations; and an open dimension problem in three dimensions. Our aims are to find strategies to deal with irregular shapes, particularly geometric tools, and solution methods for problems with unusual constraints. In the first part we look at an industrial problem related to the management of helicopter fleets. We model and test with realistic data a bin packing problem where the objective is to find the minimum aircraft needed to lift a collection of items. The characteristics of this problem allow us to relax the geometrical constraints and consider it as a variant of the one-dimensional bin packing problem, but its many problem specific constraints make this a multi-objective that, to the best of our knowledge, is new in the literature. In the second part, we deal with a novel problem in two dimensions, motivated by the deciphering of an ancient Aztec codex. The problem itself is a novel packing prob- lem with irregular shapes, an irregular container, free rotation and with the overlap and containment constraints relaxed. We provide a constructive algorithm and a meta- heuristic procedure that are able to find satisfying solutions for an open question in the deciphering of the codex. Finally, in the last part we treat three-dimensional irregular shapes. We adopt a discretised approach that allows us to generate quick intersection tests and we develop the no-fit voxel. This is an extension of the no-fit polygon, a mainstream tool for two- dimensional packing problems that had not been extended to three dimensions in the literature. Using this tool, we investigate local search neighbourhoods and metaheuristic algorithms to find efficient packings and are able to provide an ILP model based on the no-fit voxel to locally improve the packing layouts. Contents Declaration of Authorship xv Acknowledgements xvii 1 Introduction1 1.1 Objectives and contribution..........................3 1.2 Layout......................................4 2 Literature Review5 2.1 Overview of C&P problem types.......................6 2.1.1 Output maximisation.........................6 2.1.1.1 Knapsack problem......................7 2.1.1.2 Identical item packing problem...............7 2.1.1.3 Placement problem.....................7 2.1.2 Input minimisation...........................8 2.1.2.1 Bin packing problem.....................9 2.1.2.2 Cutting stock problem....................9 2.1.2.3 Open dimension problem..................9 2.1.3 Benefits of the typology........................ 10 2.2 One-dimensional literature........................... 10 2.2.1 Exact methods............................. 11 2.2.2 Approximation algorithms....................... 14 2.3 Two-dimensional literature.......................... 15 2.3.1 Regular................................. 15 2.3.2 Irregular................................. 16 2.3.2.1 Exact methods........................ 17 2.3.2.2 Metaheuristics........................ 19 2.4 Three-dimensional literature.......................... 22 2.4.1 Regular................................. 23 2.4.2 Irregular................................. 24 3 Methodology 27 3.1 Geometry representations........................... 27 3.1.1 Phi-objects............................... 28 3.1.2 Polygonal representations....................... 31 3.1.2.1 Avoiding overlap in two-dimensions............ 32 Direct trigonometry...................... 32 No-fit polygon.......................... 34 v vi CONTENTS 3.1.2.2 Avoiding overlap in three-dimensions........... 36 3.1.3 Discrete representations........................ 38 Raster methods......................... 39 Shape approximations..................... 41 3.2 Optimisation.................................. 41 3.2.1 Complexity............................... 42 3.2.2 Integer Linear Programming Models................. 43 3.2.3 Heuristics................................ 44 3.2.3.1 Constructive algorithms................... 45 3.2.3.2 Local search......................... 45 3.2.3.3 Approximation algorithms................. 46 3.2.4 Metaheuristics............................. 47 3.2.4.1 Iterated local search..................... 47 3.2.4.2 Simulated annealing..................... 48 3.2.4.3 Tabu search......................... 48 3.2.4.4 Variable neighbourhood search............... 49 3.2.4.5 Genetic algorithms...................... 49 3.2.4.6 Other metaheuristics.................... 50 3.2.5 Matheuristics.............................. 50 3.2.6 Hyper-heuristics............................ 51 3.3 Conclusion................................... 51 4 Efficient management of heterogeneous helicopter fleets 53 4.1 Introduction................................... 53 4.2 Problem description.............................. 56 Bin packing........................... 57 Placement constraints..................... 57 Objectives............................ 58 4.2.1 Bounds................................. 60 4.2.2 Exact method.............................. 61 4.2.3 Constructive.............................. 64 4.2.4 Placement rules............................. 64 4.2.5 Distance balance heuristics...................... 66 4.2.6 Genetic Algorithm........................... 66 4.2.6.1 Chromosome representation................. 67 4.2.6.2 Crossover operator...................... 67 4.2.6.3 Mutation........................... 70 4.2.6.4 Initial population...................... 70 4.2.6.5 Fitness function....................... 70 4.2.6.6 Algorithm description.................... 71 4.2.6.7 Parameters.......................... 71 4.2.7 Heterogeneous bins........................... 73 4.3 Computational experiments.......................... 74 4.3.1 Randomly generated instances.................... 75 4.3.2 Realistic instance............................ 77 4.4 Conclusions................................... 78 CONTENTS vii 5 An archaeological irregular packing problem: Packing to decipher an ancient Aztec codex 81 5.1 Introduction................................... 82 5.1.1 Historical context............................ 82 5.1.1.1 Acolh´uaarithmetic and geometry............. 83 5.1.1.2 Geographical location of the terrains............ 84 5.1.2 Problem description.......................... 84 5.1.3 Related work.............................. 87 5.2 Solution methods................................ 88 5.2.1 Constructive algorithm........................ 88 Finding potential placements................. 89 5.2.1.1 Alternative objective function............... 91 5.2.2 Local search............................... 92 5.2.3 Genetic algorithm........................... 94 5.2.3.1 Encoding & decoding.................... 95 5.2.3.2 Fitness............................ 96 5.2.3.3 Crossover........................... 96 5.2.3.4 Mutation........................... 96 5.3 Implementation and Computational results................. 96 5.3.1 Irregular strip packing instances................... 97 5.3.2 Real instance.............................. 97 5.4 Conclusion and further work......................... 99 6 Voxel-Based 3D Irregular Packing 103 6.1 Introduction................................... 104 6.2 Literature review................................ 104 6.3 Voxelised three-dimensional packing..................... 105 6.3.1 Problem description.......................... 106 6.3.2 Voxel representation.......................... 106 6.3.3 Constraint handling.......................... 106 6.4 ILP formulation................................. 109 6.5 Building blocks of the 3D packing heuristics................. 111 6.5.1 Constructive algorithm........................ 111 6.5.2 Sequence-based neighbourhoods.................... 112 Sequence swap neighbourhood................. 112 Rule change neighbourhood.................. 112 6.5.3 Layout-based neighbourhoods..................... 113 Axis aligned direction neighbourhood............. 113 Enclosing cube neighbourhood................ 114 Piece swap neighbourhood................... 114 6.5.4 Strategic Oscillation.......................... 114 6.5.5 Objective function........................... 115 6.6 Search algorithms................................ 116 6.6.1 Iterated local search.......................... 116 6.6.2 Iterated Tabu Search.......................... 117 6.6.3 Variable Neighbourhood Search.................... 119 6.7 Computational experiments.......................... 123 viii CONTENTS 6.7.1 Randomly generated instances.................... 123 6.7.2 Instances from the literature..................... 127 6.7.3 Realistic instances........................... 130 6.7.4 Discussion................................ 133 6.8 Conclusion................................... 133 7 Conclusions 135 References 139 List of Figures 2.1 Output maximisation problem types. (Adapted from W¨ascher et al. (2007))6