Closing the Gap for Pseudo-Polynomial Strip Packing

Two-dimensional packing problems are a fundamental class of optimization problems and Strip Packing is one of the most natural and famous among them. Indeed it can be defined in just one sentence: Given a set of rectangular axis parallel items and a strip with bounded width and infinite height, the objective is to find a packing of the items into the strip minimizing the packing height. We speak of pseudo-polynomial Strip Packing if we consider algorithms with pseudo-polynomial running time with respect to the width of the strip. It is known that there is no pseudo-polynomial time algorithm for Strip Packing with a ratio better than 5/4 unless P = NP. The best algorithm so far has a ratio of 4/3 + epsilon. In this paper, we close the gap between inapproximability result and currently known algorithms by presenting an algorithm with approximation ratio 5/4 + epsilon. The algorithm relies on a new structural result which is the main accomplishment of this paper. It states that each optimal solution can be transformed with bounded loss in the objective such that it has one of a polynomial number of different forms thus making the problem tractable by standard techniques, i.e., dynamic programming. To show the conceptual strength of the approach, we extend our result to other problems as well, e.g., Strip Packing with 90 degree rotations and Contiguous Moldable Task Scheduling, and present algorithms with approximation ratio 5/4 + epsilon for these problems as well

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

Operational Research and Machine Learning Applied to Transport Systems

The New Economy, environmental sustainability and global competitiveness drive inno- vations in supply chain management and transport systems. The New Economy increases the amount and types of products that can be delivered directly to homes, challenging the organisation of last-mile delivery companies. To keep up with the challenges, deliv- ery companies are continuously seeking new innovations to allow them to pack goods faster and more efficiently. Thus, the packing problem has become a crucial factor and solving this problem effectively is essential for the success of good deliveries and logistics. On land, rail transportation is known to be the most eco-friendly transport system in terms of emissions, energy consumption, land use, noise levels, and quantities of people and goods that can be moved. It is difficult to apply innovations to the rail industry due to a number of reasons: the risk aversion nature, the high level of regulations, the very high cost of infrastructure upgrades, and the natural monopoly of resources in many countries. In the UK, however, in 2018 the Department for Transport published the Joint Rail Data Action Plan, opening some rail industry datasets for researching purposes. In line with the above developments, this thesis focuses on the research of machine learning and operational research techniques in two main areas: improving packing operations for logistics and improving various operations for passenger rail. In total, the research in this thesis will make six contributions as detailed below. The first contribution is a new mathematical model and a new heuristic to solve the Multiple Heterogeneous Knapsack Problem, giving priority to smaller bins and consid- ering some important container loading constraints. This problem is interesting because many companies prefer to deal with smaller bins as they are less expensive. Moreover, giving priority to filling small bins (rather than large bins) is very important in some industries, e.g. fast-moving consumer goods. The second contribution is a novel strategy to hybridize operational research with ma- chine learning to estimate if a particular packing solution is feasible in a constant O(1) computational time. Given that traditional feasibility checking for packing solutions is an NP-Hard problem, it is expected that this strategy will significantly save time and computational effort. The third contribution is an extended mathematical model and an algorithm to apply the packing problem to improving the seat reservation system in passenger rail. The problem is formulated as the Group Seat Reservation Knapsack Problem with Price on Seat. It is an extension of the Offline Group Seat Reservation Knapsack Problem. This extension introduces a profit evaluation dependent on not only the space occupied, but also on the individual profit brought by each reserved seat. The fourth contribution is a data-driven method to infer the feasible train routing strate- gies from open data in the United Kingdom rail network. Briefly, most of the UK network is divided into sections called berths, and the transition point from one berth to another is called a berth step. There are sensors at berth steps that can detect the movement when a train passes by. The result of the method is a directed graph, the berth graph, where each node represents a berth and each arc represents a berth-step. The arcs rep- resent the feasible routing strategies, i.e. where a train can move from one berth. A connected path between two berths represents a connected section of the network. The fifth contribution is a novel method to estimate the amount of time that a train is going to spend on a berth. This chapter compares two different approaches, AutoRe- gressive Moving Average with Recurrent Neural Networks, and analyse the pros and cons of each choice with statistical analyses. The method is tested on a real-world case study, one berth that represent a busy junction in the Merseyside region. The sixth contribution is an adaptive method to forecast the running time of a train journey using the Gated Recurrent Units method. The method exploits the TD’s berth information and the berth graph. The case-study adopted in the experimental tests is the train network in the Merseyside region

Random Sequential Addition of Hard Spheres in High Euclidean Dimensions

Employing numerical and theoretical methods, we investigate the structural characteristics of random sequential addition (RSA) of congruent spheres in $d$-dimensional Euclidean space $\mathbb{R}^d$ in the infinite-time or saturation limit for the first six space dimensions ($1 \le d \le 6$). Specifically, we determine the saturation density, pair correlation function, cumulative coordination number and the structure factor in each =of these dimensions. We find that for $2 \le d \le 6$, the saturation density $\phi_s$ scales with dimension as $\phi_s= c_1/2^d+c_2 d/2^d$, where $c_1=0.202048$ and $c_2=0.973872$. We also show analytically that the same density scaling persists in the high-dimensional limit, albeit with different coefficients. A byproduct of this high-dimensional analysis is a relatively sharp lower bound on the saturation density for any $d$ given by $\phi_s \ge (d+2)(1-S_0)/2^{d+1}$, where $S_0\in [0,1]$ is the structure factor at $k=0$ (i.e., infinite-wavelength number variance) in the high-dimensional limit. Consistent with the recent "decorrelation principle," we find that pair correlations markedly diminish as the space dimension increases up to six. Our work has implications for the possible existence of disordered classical ground states for some continuous potentials in sufficiently high dimensions.Comment: 38 pages, 9 figures, 4 table

Moldable Items Packing Optimization

This research has led to the development of two mathematical models to optimize the problem of packing a hybrid mix of rigid and moldable items within a three-dimensional volume. These two developed packing models characterize moldable items from two perspectives: (1) when limited discrete configurations represent the moldable items and (2) when all continuous configurations are available to the model. This optimization scheme is a component of a lean effort that attempts to reduce the lead-time associated with the implementation of dynamic product modifications that imply packing changes. To test the developed models, they are applied to the dynamic packing changes of Meals, Ready-to-Eat (MREs) at two different levels: packing MRE food items in the menu bags and packing menu bags in the boxes. These models optimize the packing volume utilization and provide information for MRE assemblers, enabling them to preplan for packing changes in a short lead-time. The optimization results are validated by running the solutions multiple times to access the consistency of solutions. Autodesk Inventor helps visualize the solutions to communicate the optimized packing solutions with the MRE assemblers for training purposes

Optimization methods for side-chain positioning and macromolecular docking

This dissertation proposes new optimization algorithms targeting protein-protein docking which is an important class of problems in computational structural biology. The ultimate goal of docking methods is to predict the 3-dimensional structure of a stable protein-protein complex. We study two specific problems encountered in predictive docking of proteins. The first problem is Side-Chain Positioning (SCP), a central component of homology modeling and computational protein docking methods. We formulate SCP as a Maximum Weighted Independent Set (MWIS) problem on an appropriately constructed graph. Our formulation also considers the significant special structure of proteins that SCP exhibits for docking. We develop an approximate algorithm that solves a relaxation of MWIS and employ randomized estimation heuristics to obtain high-quality feasible solutions to the problem. The algorithm is fully distributed and can be implemented on multi-processor architectures. Our computational results on a benchmark set of protein complexes show that the accuracy of our approximate MWIS-based algorithm predictions is comparable with the results achieved by a state-of-the-art method that finds an exact solution to SCP. The second problem we target in this work is protein docking refinement. We propose two different methods to solve the refinement problem. The first approach is based on a Monte Carlo-Minimization (MCM) search to optimize rigid-body and side-chain conformations for binding. In particular, we study the impact of optimally positioning the side-chains in the interface region between two proteins in the process of binding. We report computational results showing that incorporating side-chain flexibility in docking provides substantial improvement in the quality of docked predictions compared to the rigid-body approaches. Further, we demonstrate that the inclusion of unbound side-chain conformers in the side-chain search introduces significant improvement in the performance of the docking refinement protocols. In the second approach, we propose a novel stochastic optimization algorithm based on Subspace Semi-Definite programming-based Underestimation (SSDU), which aims to solve protein docking and protein structure prediction. SSDU is based on underestimating the binding energy function in a permissive subspace of the space of rigid-body motions. We apply Principal Component Analysis (PCA) to determine the permissive subspace and reduce the dimensionality of the conformational search space. We consider the general class of convex polynomial underestimators, and formulate the problem of finding such underestimators as a Semi-Definite Programming (SDP) problem. Using these underestimators, we perform a biased sampling in the vicinity of the conformational regions where the energy function is at its global minimum. Moreover, we develop an exploration procedure based on density-based clustering to detect the near-native regions even when there are many local minima residing far from each other. We also incorporate a Model Selection procedure into SSDU to pick a predictive conformation. Testing our algorithm over a benchmark of protein complexes indicates that SSDU substantially improves the quality of docking refinement compared with existing methods

High Multiplicity Strip Packing

In the two-dimensional high multiplicity strip packing problem (HMSPP), we are given k distinct rectangle types, where each rectangle type Ti has ni rectangles each with width 0 \u3c wi and height 0 \u3c hi The goal is to pack these rectangles into a strip of width 1, without rotating or overlapping the rectangles, such that the total height of the packing is minimized. Let OPT(I) be the optimal height of HMSPP on input I. In this thesis, we consider HMSPP for the case when k = 3 and present an OPT(I) + 5/3 polynomial time approximation algorithm for it. Additionally, we consider HMSPP for the case when k = 4 and present an OPT(I) + 5/2 polynomial time approximation algorithm for it

Advances in flexible manipulation through the application of AI-based techniques

282 p.Objektuak hartu eta uztea oinarrizko bi eragiketa dira ia edozein aplikazio robotikotan. Gaur egun, "pick and place" aplikazioetarako erabiltzen diren robot industrialek zeregin sinpleak eta errepikakorrak egiteko duten eraginkortasuna dute ezaugarri. Hala ere, sistema horiek oso zurrunak dira, erabat kontrolatutako inguruneetan lan egiten dute, eta oso kostu handia dakarte beste zeregin batzuk egiteko birprogramatzeak. Gaur egun, industria-ingurune desberdinetako zereginak daude (adibidez, logistika-ingurune batean eskaerak prestatzea), zeinak objektuak malgutasunez manipulatzea eskatzen duten, eta oraindik ezin izan dira automatizatu beren izaera dela-eta. Automatizazioa zailtzen duten botila-lepo nagusiak manipulatu beharreko objektuen aniztasuna, roboten trebetasun falta eta kontrolatu gabeko ingurune dinamikoen ziurgabetasuna dira.Adimen artifizialak (AA) gero eta paper garrantzitsuagoa betetzen du robotikaren barruan, robotei zeregin konplexuak betetzeko beharrezko adimena ematen baitie. Gainera, AAk benetako esperientzia erabiliz portaera konplexuak ikasteko aukera ematen du, programazioaren kostua nabarmen murriztuz. Objektuak manipulatzeko egungo sistema robotikoen mugak ikusita, lan honen helburu nagusia manipulazio-sistemen malgutasuna handitzea da AAn oinarritutako algoritmoak erabiliz, birprogramatu beharrik gabe ingurune dinamikoetara egokitzeko beharrezko gaitasunak emanez