Optimizing a multiple objective surgical case scheduling problem.

The scheduling of the operating theater on a daily base is a complicated task and is mainly based on the experience of the human planner. This, however, does not mean that this task can be seen as unimportant since the schedule of individual surgeries influences a medical department as a whole. Based on practical suggestions of the planner and on real-life constraints, we will formulate a multiple objective optimization model in order to facilitate this decision process. We will show that this optimization problem is NP-hard and hence hard to solve. Both exact and heuristic algorithms, based on integer programming and on implicit enumeration (branch-and-bound), will be introduced. These solution approaches will be thoroughly tested on a realistic test set using data of the surgical day-care center at the university hospital Gasthuisberg in Leuven (Belgium). Finally, results will be analyzed and conclusions will be formulated.Algorithms; Belgium; Branch-and-bound; Constraint; Data; Decision; Experience; Healthcare; Heuristic; Integer; Integer programming; Model; Optimization; Order; Processes; Real life; Scheduling; University;

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

Certifying Correctness for Combinatorial Algorithms : by Using Pseudo-Boolean Reasoning

Over the last decades, dramatic improvements in combinatorialoptimisation algorithms have significantly impacted artificialintelligence, operations research, and other areas. These advances,however, are achieved through highly sophisticated algorithms that aredifficult to verify and prone to implementation errors that can causeincorrect results. A promising approach to detect wrong results is touse certifying algorithms that produce not only the desired output butalso a certificate or proof of correctness of the output. An externaltool can then verify the proof to determine that the given answer isvalid. In the Boolean satisfiability (SAT) community, this concept iswell established in the form of proof logging, which has become thestandard solution for generating trustworthy outputs. The problem isthat there are still some SAT solving techniques for which prooflogging is challenging and not yet used in practice. Additionally,there are many formalisms more expressive than SAT, such as constraintprogramming, various graph problems and maximum satisfiability(MaxSAT), for which efficient proof logging is out of reach forstate-of-the-art techniques.This work develops a new proof system building on the cutting planesproof system and operating on pseudo-Boolean constraints (0-1 linearinequalities). We explain how such machine-verifiable proofs can becreated for various problems, including parity reasoning, symmetry anddominance breaking, constraint programming, subgraph isomorphism andmaximum common subgraph problems, and pseudo-Boolean problems. Weimplement and evaluate the resulting algorithms and a verifier for theproof format, demonstrating that the approach is practical for a widerange of problems. We are optimistic that the proposed proof system issuitable for designing certifying variants of algorithms inpseudo-Boolean optimisation, MaxSAT and beyond

The bus sightseeing problem

The basic characteristic of vehicle routing problems with profits (VRPP) is that locations to be visited are not predetermined. On the contrary, they are selected in pursuit of maximizing the profit collected from them. Significant research focus has been directed toward profitable routing variants due to the practical importance of their applications and their interesting structure, which jointly optimizes node selection and routing decisions. Profitable routing applications arise in the tourism industry aiming to maximize the profit score of attractions visited within a limited time period. In this paper, a new VRPP is introduced, referred to as the bus sightseeing problem (BSP). The BSP calls for determining bus tours for transporting different groups of tourists with different preferences on touristic attractions. Two interconnected decision levels have to be jointly tackled: assignment of tourists to buses and routing of buses to the various attractions. A mixed-integer programming formulation for the BSP is provided and solved by a Benders decomposition algorithm. For large-scale instances, an iterated local search based metaheuristic algorithm is developed with some tailored neighborhood operators. The proposed methods are tested on a large family of test instances, and the obtained computational results demonstrate the effectiveness of the proposed solution approaches

Models and algorithms for decomposition problems

This thesis deals with the decomposition both as a solution method and as a problem itself. A decomposition approach can be very effective for mathematical problems presenting a specific structure in which the associated matrix of coefficients is sparse and it is diagonalizable in blocks. But, this kind of structure may not be evident from the most natural formulation of the problem. Thus, its coefficient matrix may be preprocessed by solving a structure detection problem in order to understand if a decomposition method can successfully be applied. So, this thesis deals with the k-Vertex Cut problem, that is the problem of finding the minimum subset of nodes whose removal disconnects a graph into at least k components, and it models relevant applications in matrix decomposition for solving systems of equations by parallel computing. The capacitated k-Vertex Separator problem, instead, asks to find a subset of vertices of minimum cardinality the deletion of which disconnects a given graph in at most k shores and the size of each shore must not be larger than a given capacity value. Also this problem is of great importance for matrix decomposition algorithms. This thesis also addresses the Chance-Constrained Mathematical Program that represents a significant example in which decomposition techniques can be successfully applied. This is a class of stochastic optimization problems in which the feasible region depends on the realization of a random variable and the solution must optimize a given objective function while belonging to the feasible region with a probability that must be above a given value. In this thesis, a decomposition approach for this problem is introduced. The thesis also addresses the Fractional Knapsack Problem with Penalties, a variant of the knapsack problem in which items can be split at the expense of a penalty depending on the fractional quantity

Decomposition-Based Integer Programming, Stochastic Programming, and Robust Optimization Methods for Healthcare Planning, Scheduling, and Routing Problems

RÉSUMÉ : Il existe de nombreuses applications de planification, d’ordonnancement et de confection de tournées dans les systèmes de santé. La résolution efficace de ces problèmes peut aider les responsables de la santé à fournir des services de meilleure qualité, en utilisant efficacement les ressources médicales disponibles. En raison de la nature combinatoire de ces problèmes, dans de nombreux cas, les algorithmes de programmation en nombres entiers standards dans les logiciels commerciaux de programmation mathématique tels que CPLEX et Gurobi ne peuvent pas résoudre efficacement les modèles correspondants. Dans cette thèse, nous étudions trois problèmes de planification, d’ordonnancement et de confection de tournées des soins de santé et proposons des approches à base de décomposition utilisant la programmation en nombres entiers, la programmation stochastique et une méthode d’optimisation robuste. Le premier article de cette thèse présente un problème intégré de planification et d’ordonnancement dans le cadre des salles d’opération. Cette situation implique d’optimiser l’ordonnancement et l’affectation des chirurgies aux différentes salles d’opération, sur un horizon de planification à court terme. Nous avons pris en compte les heures de travail quotidiennes maximales des chirurgiens, le temps de nettoyage obligatoire alloué lors du passage de cas infectieux à des cas non infectieux et le respect des dates limites des chirurgies. Nous avons aussi empêché le chevauchement des chirurgies effectuées par le même chirurgien. Nous avons formulé le problème en utilisant un modèle de programmation mathématique et développé un algorithme «branch-and-price-and-cut» basé sur un modèle de programmation par contraintes pour le sous-problème. Nous avons mis en place des règles de dominance et un algorithme de détection d’infaillibilité rapide. Cet algorithme, basé sur le problème du sac à dos multidimensionnel, nous permet d’améliorer l’efficacité du modèle de programmation de contraintes. Les résultats montrent que notre méthode présente un écart à l’optimum moyen de 2,81%, ce qui surpasse de manière significative la formulation mathématique compacte dans la littérature. Dans la deuxième partie de cette thèse, pour la première fois, nous avons étudié l’optimisation des problèmes de tournées de véhicules avec visites synchronisées (VRPS) en tenant compte de stochasticité des temps de déplacement et de service. En plus d’envisager un problème d’ordonnancement des soins de santé à domicile, nous introduisons un problème d’ordonnancement des salles d’opération avec des durées stochastiques qui est une nouvelle application de VRPS. Nous avons modélisé les VRPS qui ont des durées stochastiques en programmation stochastique à deux niveaux avec des variables entières dans les deux niveaux. L’avantage du modèle proposé est que, contrairement aux modèles déterministes de la littérature VRPS, il n’a pas de contraintes «big-M». Cet avantage entraine en contrepartie la présence d’un grand nombre de variables entières dans le second niveau. Nous avons prouvé que les contraintes d’intégralité sur les variables du deuxième niveau sont triviales ce qui nous permet d’appliquer l’algorithme «L-shaped» et son implémentation branch-and-and-cut pour résoudre le problème. Nous avons amélioré le modèle en développant des inégalités valides et une fonction de bornes inférieures. Nous avons analysé les sous-problèmes de l’algorithme en L et nous avons proposé une méthode de résolution qui est beaucoup plus rapide que les algorithmes de programmation linéaire standards. En outre, nous avons étendu notre modèle pour modéliser les VRPS avec des temps de déplacement et de service dépendant du temps. Les résultats de l’optimisation montrent que, pour le problème stochastique de soins à domicile, l’algorithme «branch-and-cut» résout à l’optimalité les exemplaires avec 15 patients et 10% à 30% de visites synchronisées. Il trouve également des solutions avec un écart à l’optimum moyen de de 3,57% pour les cas avec 20 patients. De plus l’algorithme «branch-and-cut» résout à l’optimalité les problèmes d’ordonnancement stochastique des salles d’opération avec 20 chirurgies. Ceci est une amélioration considérable par rapport à la littérature qui fait état de cas avec 11 chirurgies. En outre, la modélisation proposée pour le problème dépendant du temps trouve des solutions optimales pour d’une grande portion des exemplaires d’ordonnancement de soins de santé à domicile avec 30 à 60 patients et différents taux de visites synchronisées. Dans la dernière partie de cette thèse, nous avons étudié une catégorie de modèles d’optimisation robuste en deux étapes avec des variables entières du problème adversaire. Nous avons analysé l’importance de cette classe de problèmes lors de la modélisation à deux niveaux de problèmes de planification de ressources robuste en deux étapes où certaines tâches ont des temps d’arrivée et des durées incertains. Nous considérons un problème de répartition et d’affectation d’infirmières comme une application de cette classe de modèles robustes. Nous avons appliqué la décomposition de Dantzig-Wolfe pour exploiter la structure de ces modèles, ce qui nous a permis de montrer que le problème initial se réduit à un problème robuste à une seule étape. Nous avons proposé un algorithme Benders pour le problème reformulé. Étant donné que le problème principal et le sous-problème dans l’algorithme Benders sont des programmes à nombres entiers mixtes, il requiert une quantité de calcul importante à chaque itération de l’algorithme pour les résoudre de manière optimale. Par conséquent, nous avons développé de nouvelles conditions d’arrêt pour ces programmes à nombres entiers mixtes et fourni des preuves de convergence. Nous avons développé également un algorithme heuristique appelé «dual algorithm». Dans cette heuristique, nous dualisons la relaxation linéaire du problème adversaire dans le problème reformulé et générons des coupes itérativement pour façonner l’enveloppe convexe de l’ensemble d’incertitude. Nous avons combiné cette heuristique avec l’algorithme Benders pour créer un algorithme plus efficace appelé algorithme «Benders-dual algorithm». De nombreuses expériences de calcul sur le problème de répartition et d’affectation d’infirmières sont effectuées pour comparer ces algorithmes.----------ABSTRACT : There are many applications of planning, scheduling, and routing problems in healthcare systems. Efficiently solving these problems can help healthcare managers provide higher-quality services by making efficient use of available medical resources. Because of the combinatorial nature of these problems, in many cases, standard integer programming algorithms in commercial mathematical programming software such as CPLEX and Gurobi cannot solve the corresponding models effectively. In this dissertation, we study three healthcare planning, scheduling, and routing problems and propose decomposition-based integer programming, stochastic programming, and robust optimization methods for them. In the first essay of this dissertation, we study an integrated operating room planning and scheduling problem that combines the assignment of surgeries to operating rooms and scheduling over a short-term planning horizon. We take into account the maximum daily working hours of surgeons, prevent the overlapping of surgeries performed by the same surgeon, allow time for the obligatory cleaning when switching from infectious to noninfectious cases, and respect the surgery deadlines. We formulate the problem using a mathematical programming model and develop a branch-and-price-and-cut algorithm based on a constraint programming model for the subproblem. We also develop dominance rules and a fast infeasibility-detection algorithm based on a multidimensional knapsack problem to improve the efficiency of the constraint programming model. The computational results show that our method has an average optimality gap of 2.81% and significantly outperforms a compact mathematical formulation in the literature. As the second essay of this dissertation, for the first time, we study vehicle routing problems with synchronized visits (VRPS) and stochastic/time-dependent travel and service times. In addition to considering a home-health care scheduling problem, we introduce an operating room scheduling problem with stochastic durations as a novel application of VRPS. We formulate VRPS with stochastic times as a two-stage stochastic programming model with integer variables in both stages. An advantage of the proposed model is that, in contrast to the deterministic models in the VRPS literature, it does not have any big-M constraints. This advantage comes at the cost of a large number of second-stage integer variables. We prove that the integrality constraints on second-stage variables are trivial, and therefore we can apply the L-shaped algorithm and its branch-and-cut implementation to solve the problem. We enhance the model by developing valid inequalities and a lower bounding functional. We analyze the subproblems of the L-shaped algorithm and devise a solution method for them that is much faster than standard linear programming algorithms. Moreover, we extend our model to formulate VRPS with time-dependent travel and service times. Computational results show that, in the stochastic home-health care scheduling problem, the branch-and-cut algorithm optimally solves instances with 15 patients and 10% to 30% of synchronized visits. It also finds solutions with an average optimality gap of 3.57% for instances with 20 patients. Furthermore, the branch-and-cut algorithm ptimally solves stochastic operating room scheduling problems with 20 surgeries, a considerable improvement over the literature that reports on instances with 11 surgeries. In addition, the proposed formulation for the time-dependent problem solves a large portion of home-health care scheduling instances with 30 to 60 patients and different rates of synchronized visits to optimality. For the last essay of this dissertation, we also study a class of two-stage robust optimization models with integer adversarial variables. We discuss the importance of this class of problems in modeling two-stage robust resource planning problems where some tasks have uncertain arrival times and duration periods. We consider a two-stage nurse planning problem as an application of this class of robust models. We apply Dantzig-Wolfe decomposition to exploit the structure of these models and show that the original problem reduces to a singlestage robust problem. We propose a Benders algorithm for the reformulated single-stage problem. Since the master problem and subproblem in the Benders algorithm are mixed integer programs, it is computationally demanding to solve them optimally at each iteration of the algorithm. Therefore, we develop novel stopping conditions for these mixed integer programs and provide the relevant convergence proofs. We also develop a heuristic algorithm called dual algorithm. In this heuristic, we dualize the linear programming relaxation of the adversarial problem in the reformulated problem and iteratively generate cuts to shape the convex hull of the uncertainty set. We combine this heuristic with the Benders algorithm to create a more effective algorithm called Benders-dual algorithm. Extensive computational experiments on the nurse planning problem are performed to compare these algorithms