Optimization via Benders' Decomposition

In a period when optimization has entered almost every facet of our lives, this thesis is designed to establish an understanding about the rather contemporary optimization technique: Benders' Decomposition. It can be roughly stated as a method that handles problems with complicating variables, which when temporarily fixed, yield a problem much easier to solve. We examine the classical Benders' Decomposition algorithm in greater depth followed by a mathematical defense to verify the correctness, state how the convergence of the algorithm depends on the formulation of the problem, identify its correlation to other well-known decomposition methods for Linear Programming problems, and discuss some real-world examples. We introduce present extensions of the method that allow its application to a wider range of problems. We also present a classification of acceleration strategies which is centered round the key sections of the algorithm. We conclude by illustrating the shortcomings, trends, and potential research directions

Branching strategies for mixed-integer programs containing logical constraints and decomposable structure

Decision-making optimisation problems can include discrete selections, e.g. selecting a route, arranging non-overlapping items or designing a network of items. Branch-and-bound (B&B), a widely applied divide-and-conquer framework, often solves such problems by considering a continuous approximation, e.g. replacing discrete variable domains by a continuous superset. Such approximations weaken the logical relations, e.g. for discrete variables corresponding to Boolean variables. Branching in B&B reintroduces logical relations by dividing the search space. This thesis studies designing B&B branching strategies, i.e. how to divide the search space, for optimisation problems that contain both a logical and a continuous structure. We begin our study with a large-scale, industrially-relevant optimisation problem where the objective consists of machine-learnt gradient-boosted trees (GBTs) and convex penalty functions. GBT functions contain if-then queries which introduces a logical structure to this problem. We propose decomposition-based rigorous bounding strategies and an iterative heuristic that can be embedded into a B&B algorithm. We approach branching with two strategies: a pseudocost initialisation and strong branching that target the structure of GBT and convex penalty aspects of the optimisation objective, respectively. Computational tests show that our B&B approach outperforms state-of-the-art solvers in deriving rigorous bounds on optimality. Our second project investigates how satisfiability modulo theories (SMT) derived unsatisfiable cores may be utilised in a B&B context. Unsatisfiable cores are subsets of constraints that explain an infeasible result. We study two-dimensional bin packing (2BP) and develop a B&B algorithm that branches on SMT unsatisfiable cores. We use the unsatisfiable cores to derive cuts that break 2BP symmetries. Computational results show that our B&B algorithm solves 20% more instances when compared with commercial solvers on the tested instances. Finally, we study convex generalized disjunctive programming (GDP), a framework that supports logical variables and operators. Convex GDP includes disjunctions of mathematical constraints, which motivate branching by partitioning the disjunctions. We investigate separation by branching, i.e. eliminating solutions that prevent rigorous bound improvement, and propose a greedy algorithm for building the branches. We propose three scoring methods for selecting the next branching disjunction. We also analyse how to leverage infeasibility to expedite the B&B search. Computational results show that our scoring methods can reduce the number of explored B&B nodes by an order of magnitude when compared with scoring methods proposed in literature. Our infeasibility analysis further reduces the number of explored nodes.Open Acces

Managing management innovations: Contextual complexity and the pursuit of improvements in healthcare

In a context characterised by complexity and conflicting demands, healthcare managers at a meso-level struggle to pursue improvements in the quality and efficiency of care operations. An influential approach on how to pursue improvements is quality management (QM). QM adopts the view that systems are centred around a common aim and should be appreciated and managed to reduce undesired variation and improve performance incrementally. Nuancing this view, complexity science propels the idea of healthcare as a complex adaptive system (CAS), which refutes prediction and managerial control of development. As one component of the CAS of healthcare, various management innovations (MIs) provide suggestions on how to achieve improvements. However, achieving any improvement is not often as simple as portrayed and MIs can rarely be fully and exclusively applied in practice. Starting from the practical issue of how to achieve improvements in healthcare, this thesis seeks to explore how healthcare managers at a meso-level can understand and use MIs to handle complexity and achieve improvements. A qualitative and action research-inspired approach is adopted to investigate this issue, concentrating on the context of psychiatric care at the Sahlgrenska University Hospital in Gothenburg, Sweden.Four studies, resulting in five appended papers, are presented. By investigating contemporary MIs, the studies contribute to an improved understanding of how MIs can be used, and complexity handled, in the pursuit of improvements. Study 1 starts by exploring the concept of value at a time when lean was succeeded by value-based healthcare (VBHC) as the MI in fashion in the context and the study follows the implementation of VBHC in an action research-inspired approach. Study 2 tests the utility of the value configurations framework to handle conflicting logics and pursue improvements in psychosis care. In study 3, literature on network configurations in different healthcare contexts is reviewed. Lastly, study 4 is an action research study focusing on contextualisation of learning health systems (LHS) as yet an example of an MI in healthcare. Based on the findings of five appended papers and earlier literature from the fields of QM, complexity science, and MIs, a model is developed that points to the centrality and utility of logics to connect MIs and other system components to improve the understanding of both MIs and CASs. By investigating the logics underlying different MIs, actors in the healthcare system (e.g., politicians, physicians, and managers), and technical features of care (e.g., its predictability and inclination to standardised treatments), a relative appreciation of a CAS can be pursued, which can guide managers in how to use MIs and attract change that can lead to improvements. Furthermore, the thesis supports the view that MIs are often ambiguous concepts that can be translated and adapted to fit a local context in a process of contextualisation. For scholars, the thesis also contributes by integrating the perspectives of QM and complexity science, and of QM and MIs in general, as two parallel approaches to pursue improvements in healthcare

Systems Analysis by Multilevel Methods: With Applications to Economics and Management

This book presents a survey of usable multilevel methods for modeling and solving decision problems in economics and management. The methods are largely extensions of linear programming and fall within the realm of column generation and decomposition. About one third of the book is concerned with methods and the rest describes case studies where these methods have actually been used. They are taken from areas such as national and regional economic planning, production planning, and transportation planning

Traffic prediction and bilevel network design

Cette thèse porte sur la modélisation du trafic dans les réseaux routiers et comment celle-ci est intégrée dans des modèles d'optimisation. Ces deux sujets ont évolué de manière plutôt disjointe: le trafic est prédit par des modèles mathématiques de plus en plus complexes, mais ce progrès n'a pas été incorporé dans les modèles de design de réseau dans lesquels les usagers de la route jouent un rôle crucial. Le but de cet ouvrage est d'intégrer des modèles d'utilités aléatoires calibrés avec de vraies données dans certains modèles biniveaux d'optimisation et ce, par une décomposition de Benders efficace. Cette décomposition particulière s'avère être généralisable par rapport à une grande classe de problèmes communs dans la litérature et permet d'en résoudre des exemples de grande taille. Le premier article présente une méthodologie générale pour utiliser des données GPS d'une flotte de véhicules afin d'estimer les paramètres d'un modèle de demande dit recursive logit. Les traces GPS sont d'abord associées aux liens d'un réseau à l'aide d'un algorithme tenant compte de plusieurs facteurs. Les chemins formés par ces suites de liens et leurs caractéristiques sont utilisés afin d'estimer les paramètres d'un modèle de choix. Ces paramètres représentent la perception qu'ont les usagers de chacune de ces caractéristiques par rapport au choix de leur chemin. Les données utilisées dans cet article proviennent des véhicules appartenant à plusieurs compagnies de transport opérant principalement dans la région de Montréal. Le deuxième article aborde l'intégration d'un modèle de choix de chemin avec utilités aléatoires dans une nouvelle formulation biniveau pour le problème de capture de flot de trafic. Le modèle proposé permet de représenter différents comportements des usagers par rapport à leur choix de chemin en définissant les utilités d'arcs appropriées. Ces utilités sont stochastiques ce qui contribue d'autant plus à capturer un comportement réaliste des usagers. Le modèle biniveau est rendu linéaire à travers l'ajout d'un terme lagrangien basé sur la dualité forte et ceci mène à une décomposition de Benders particulièrement efficace. Les expériences numériques sont principalement menés sur un réseau représentant la ville de Winnipeg ce qui démontre la possibilité de résoudre des problèmes de taille relativement grande. Le troisième article démontre que l'approche du second article peut s'appliquer à une forme particulière de modèles biniveaux qui comprennent plusieurs problèmes différents. La décomposition est d'abord présentée dans un cadre général, puis dans un contexte où le second niveau du modèle biniveau est un problème de plus courts chemins. Afin d'établir que ce contexte inclut plusieurs applications, deux applications distinctes sont adaptées à la forme requise: le transport de matières dangeureuses et la capture de flot de trafic déterministe. Une troisième application, la conception et l'établissement de prix de réseau simultanés, est aussi présentée de manière similaire à l'Annexe B de cette thèse.The subject of this thesis is the modeling of traffic in road networks and its integration in optimization models. In the literature, these two topics have to a large extent evolved independently: traffic is predicted more accurately by increasingly complex mathematical models, but this progress has not been incorporated in network design models where road users play a crucial role. The goal of this work is to integrate random utility models calibrated with real data into bilevel optimization models through an efficient Benders decomposition. This particular decomposition generalizes to a wide class of problems commonly found in the literature and can be used to solved large-scale instances. The first article presents a general methodology to use GPS data gathered from a fleet of vehicles to estimate the parameters of a recursive logit demand model. The GPS traces are first matched to the arcs of a network through an algorithm taking into account various factors. The paths resulting from these sequences of arcs, along with their characteristics, are used to estimate parameters of a choice model. The parameters represent users' perception of each of these characteristics in regards to their path choice behaviour. The data used in this article comes from trucks used by a number of transportation companies operating mainly in the Montreal region. The second article addresses the integration of a random utility maximization model in a new bilevel formulation for the general flow capture problem. The proposed model allows for a representation of different user behaviors in regards to their path choice by defining appropriate arc utilities. These arc utilities are stochastic which further contributes in capturing real user behavior. This bilevel model is linearized through the inclusion of a Lagrangian term based on strong duality which paves the way for a particularly efficient Benders decomposition. The numerical experiments are mostly conducted on a network representing the city of Winnipeg which demonstrates the ability to solve problems of a relatively large size. The third article illustrates how the approach used in the second article can be generalized to a particular form of bilevel models which encompasses many different problems. The decomposition is first presented in a general setting and subsequently in a context where the lower level of the bilevel model is a shortest path problem. In order to demonstrate that this form is general, two distinct applications are adapted to fit the required form: hazmat transportation network design and general flow capture. A third application, joint network design and pricing, is also similarly explored in Appendix B of this thesis