Optimization Models and Algorithms for Spatial Scheduling

Spatial scheduling problems involve scheduling a set of activities or jobs that each require a certain amount of physical space in order to be carried out. In these problems space is a limited resource, and the job locations, orientations, and start times must be simultaneously determined. As a result, spatial scheduling problems are a particularly difficult class of scheduling problems. These problems are commonly encountered in diverse industries including shipbuilding, aircraft assembly, and supply chain management. Despite its importance, there is a relatively scarce amount of research in the area of spatial scheduling. In this dissertation, spatial scheduling problems are studied from a mathematical and algorithmic perspective. Optimization models based on integer programming are developed for several classes of spatial scheduling problems. While the majority of these models address problems having an objective of minimizing total tardiness, the models are shown to contain a core set of constraints that are common to most spatial scheduling problems. As a result, these constraints form the basis of the models given in this dissertation and many other spatial scheduling problems with different objectives as well. The complexity of these models is shown to be at least NP-complete, and spatial scheduling problems in general are shown to be NP-hard. A lower bound for the total tardiness objective is shown, and a polynomial-time algorithm for computing this lower bound is given. The computational complexity inherent to spatial scheduling generally prevents the use of optimization models to find solutions to larger, realistic problems in a reasonable time. Accordingly, two classes of approximation algorithms were developed: greedy heuristics for finding fast, feasible solutions; and hybrid meta-heuristic algorithms to search for near-optimal solutions. A flexible hybrid algorithm framework was developed, and a number of hybrid algorithms were devised from this framework that employ local search and several varieties of simulated annealing. Extensive computational experiments showed these hybrid meta-heuristic algorithms to be effective in finding high-quality solutions over a wide variety of problems. Hybrid algorithms based on local search generally provided both the best-quality solutions and the greatest consistency

Algorithms and data structures for three-dimensional packing

Cutting and packing problems are increasingly prevalent in industry. A well utilised freight vehicle will save a business money when delivering goods, as well as reducing the environmental impact, when compared to sending out two lesser-utilised freight vehicles. A cutting machine that generates less wasted material will have a similar effect. Industry reliance on automating these processes and improving productivity is increasing year-on-year. This thesis presents a number of methods for generating high quality solutions for these cutting and packing challenges. It does so in a number of ways. A fast, efficient framework for heuristically generating solutions to large problems is presented, and a method of incrementally improving these solutions over time is implemented and shown to produce even higher packing utilisations. The results from these findings provide the best known results for 28 out of 35 problems from the literature. This framework is analysed and its effectiveness shown over a number of datasets, along with a discussion of its theoretical suitability for higher-dimensional packing problems. A way of automatically generating new heuristics for this framework that can be problem specific, and therefore highly tuned to a given dataset, is then demonstrated and shown to perform well when compared to the expert-designed packing heuristics. Finally some mathematical models which can guarantee the optimality of packings for small datasets are given, and the (in)effectiveness of these techniques discussed. The models are then strengthened and a novel model presented which can handle much larger problems under certain conditions. The thesis finishes with a discussion about the applicability of the different approaches taken to the real-world problems that motivate them

Decomposing and packing polygons / Dania el-Khechen.

In this thesis, we study three different problems in the field of computational geometry: the partitioning of a simple polygon into two congruent components, the partitioning of squares and rectangles into equal area components while minimizing the perimeter of the cuts, and the packing of the maximum number of squares in an orthogonal polygon. To solve the first problem, we present three polynomial time algorithms which given a simple polygon P partitions it, if possible, into two congruent and possibly nonsimple components P 1 and P 2 : an O ( n 2 log n ) time algorithm for properly congruent components and an O ( n 3 ) time algorithm for mirror congruent components. In our analysis of the second problem, we experimentally find new bounds on the optimal partitions of squares and rectangles into equal area components. The visualization of the best determined solutions allows us to conjecture some characteristics of a class of optimal solutions. Finally, for the third problem, we present three linear time algorithms for packing the maximum number of unit squares in three subclasses of orthogonal polygons: the staircase polygons, the pyramids and Manhattan skyline polygons. We also study a special case of the problem where the given orthogonal polygon has vertices with integer coordinates and the squares to pack are (2 {604} 2) squares. We model the latter problem with a binary integer program and we develop a system that produces and visualizes optimal solutions. The observation of such solutions aided us in proving some characteristics of a class of optimal solutions

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

A Polyhedral Study of Mixed 0-1 Set

We consider a variant of the well-known single node fixed charge network flow set with constant capacities. This set arises from the relaxation of more general mixed integer sets such as lot-sizing problems with multiple suppliers. We provide a complete polyhedral characterization of the convex hull of the given set