Combinatorial and Robust Optimisation Models and Algorithms for Railway Applications

This thesis deals with an investigation of combinatorial and robust optimisation models to solve railway problems. Railway applications represent a challenging area for operations research. In fact, most problems in this context can be modelled as combinatorial optimisation problems, in which the number of feasible solutions is finite. Yet, despite the astonishing success in the field of combinatorial optimisation, the current state of algorithmic research faces severe difficulties with highly-complex and data-intensive applications such as those dealing with optimisation issues in large-scale transportation networks. One of the main issues concerns imperfect information. The idea of Robust Optimisation, as a way to represent and handle mathematically systems with not precisely known data, dates back to 1970s. Unfortunately, none of those techniques proved to be successfully applicable in one of the most complex and largest in scale (transportation) settings: that of railway systems. Railway optimisation deals with planning and scheduling problems over several time horizons. Disturbances are inevitable and severely affect the planning process. Here we focus on two compelling aspects of planning: robust planning and online (real-time) planning.

Enhanced Mixed Integer Programming Techniques and Routing Problems

Mixed integer programming is up today one of the most widely used techniques for dealing with hard optimization problems. On the one side, many practical optimization problems arising from real-world applications (such as, e.g., scheduling, project planning, transportation, telecommunications, economics and finance, timetabling, etc) can be easily and effectively formulated as Mixed Integer linear Programs (MIPs). On the other hand, 50 and more years of intensive research has dramatically improved on the capability of the current generation of MIP solvers to tackle hard problems in practice. However, many questions are still open and not fully understood, and the mixed integer programming community is still more than active in trying to answer some of these questions. As a consequence, a huge number of papers are continuously developed and new intriguing questions arise every year. When dealing with MIPs, we have to distinguish between two different scenarios. The first one happens when we are asked to handle a general MIP and we cannot assume any special structure for the given problem. In this case, a Linear Programming (LP) relaxation and some integrality requirements are all we have for tackling the problem, and we are ``forced" to use some general purpose techniques. The second one happens when mixed integer programming is used to address a somehow structured problem. In this context, polyhedral analysis and other theoretical and practical considerations are typically exploited to devise some special purpose techniques. This thesis tries to give some insights in both the above mentioned situations. The first part of the work is focused on general purpose cutting planes, which are probably the key ingredient behind the success of the current generation of MIP solvers. Chapter 1 presents a quick overview of the main ingredients of a branch-and-cut algorithm, while Chapter 2 recalls some results from the literature in the context of disjunctive cuts and their connections with Gomory mixed integer cuts. Chapter 3 presents a theoretical and computational investigation of disjunctive cuts. In particular, we analyze the connections between different normalization conditions (i.e., conditions to truncate the cone associated with disjunctive cutting planes) and other crucial aspects as cut rank, cut density and cut strength. We give a theoretical characterization of weak rays of the disjunctive cone that lead to dominated cuts, and propose a practical method to possibly strengthen those cuts arising from such weak extremal solution. Further, we point out how redundant constraints can affect the quality of the generated disjunctive cuts, and discuss possible ways to cope with them. Finally, Chapter 4 presents some preliminary ideas in the context of multiple-row cuts. Very recently, a series of papers have brought the attention to the possibility of generating cuts using more than one row of the simplex tableau at a time. Several interesting theoretical results have been presented in this direction, often revisiting and recalling other important results discovered more than 40 years ago. However, is not clear at all how these results can be exploited in practice. As stated, the chapter is a still work-in-progress and simply presents a possible way for generating two-row cuts from the simplex tableau arising from lattice-free triangles and some preliminary computational results. The second part of the thesis is instead focused on the heuristic and exact exploitation of integer programming techniques for hard combinatorial optimization problems in the context of routing applications. Chapters 5 and 6 present an integer linear programming local search algorithm for Vehicle Routing Problems (VRPs). The overall procedure follows a general destroy-and-repair paradigm (i.e., the current solution is first randomly destroyed and then repaired in the attempt of finding a new improved solution) where a class of exponential neighborhoods are iteratively explored by heuristically solving an integer programming formulation through a general purpose MIP solver. Chapters 7 and 8 deal with exact branch-and-cut methods. Chapter 7 presents an extended formulation for the Traveling Salesman Problem with Time Windows (TSPTW), a generalization of the well known TSP where each node must be visited within a given time window. The polyhedral approaches proposed for this problem in the literature typically follow the one which has been proven to be extremely effective in the classical TSP context. Here we present an overall (quite) general idea which is based on a relaxed discretization of time windows. Such an idea leads to a stronger formulation and to stronger valid inequalities which are then separated within the classical branch-and-cut framework. Finally, Chapter 8 addresses the branch-and-cut in the context of Generalized Minimum Spanning Tree Problems (GMSTPs) (i.e., a class of NP-hard generalizations of the classical minimum spanning tree problem). In this chapter, we show how some basic ideas (and, in particular, the usage of general purpose cutting planes) can be useful to improve on branch-and-cut methods proposed in the literature.

Decomposition and reformulation of integer linear programming problems

This thesis deals with an investigation of Decomposition and Reformulation to solve Integer Linear Programming Problems. This method is often a very successful approach computationally, producing high-quality solutions for well-structured combinatorial optimization problems like vehicle routing, cutting stock, p-median and generalized assignment . However, until now the method has always been tailored to the specific problem under investigation. The principal innovation of this thesis is to develop a new framework able to apply this concept to a generic MIP problem. The new approach is thus capable of auto-decomposition and autoreformulation of the input problem applicable as a resolving black box algorithm and works as a complement and alternative to the normal resolving techniques. The idea of Decomposing and Reformulating (usually called in literature Dantzig and Wolfe Decomposition DWD) is, given a MIP, to convexify one (or more) subset(s) of constraints (slaves) and working on the partially convexified polyhedron(s) obtained. For a given MIP several decompositions can be defined depending from what sets of constraints we want to convexify. In this thesis we mainly reformulate MIPs using two sets of variables: the original variables and the extended variables (representing the exponential extreme points). The master constraints consist of the original constraints not included in any slaves plus the convexity constraint(s) and the linking constraints(ensuring that each original variable can be viewed as linear combination of extreme points of the slaves). The solution procedure consists of iteratively solving the reformulated MIP (master) and checking (pricing) if a variable of reduced costs exists, and in which case adding it to the master and solving it again (columns generation), or otherwise stopping the procedure. The advantage of using DWD is that the reformulated relaxation gives bounds stronger than the original LP relaxation, in addition it can be incorporated in a Branch and bound scheme (Branch and Price) in order to solve the problem to optimality. If the computational time for the pricing problem is reasonable this leads in practice to a stronger speed up in the solution time, specially when the convex hull of the slaves is easy to compute, usually because of its special structure.

Hybrid constraint programming and metaheuristic methods for large scale optimization problems

This work presents hybrid Constraint Programming (CP) and metaheuristic methods for the solution of Large Scale Optimization Problems; it aims at integrating concepts and mechanisms from the metaheuristic methods to a CP-based tree search environment in order to exploit the advantages of both approaches. The modeling and solution of large scale combinatorial optimization problem is a topic which has arisen the interest of many researcherers in the Operations Research field; combinatorial optimization problems are widely spread in everyday life and the need of solving difficult problems is more and more urgent. Metaheuristic techniques have been developed in the last decades to effectively handle the approximate solution of combinatorial optimization problems; we will examine metaheuristics in detail, focusing on the common aspects of different techniques. Each metaheuristic approach possesses its own peculiarities in designing and guiding the solution process; our work aims at recognizing components which can be extracted from metaheuristic methods and re-used in different contexts. In particular we focus on the possibility of porting metaheuristic elements to constraint programming based environments, as constraint programming is able to deal with feasibility issues of optimization problems in a very effective manner. Moreover, CP offers a general paradigm which allows to easily model any type of problem and solve it with a problem-independent framework, differently from local search and metaheuristic methods which are highly problem specific. In this work we describe the implementation of the Local Branching framework, originally developed for Mixed Integer Programming, in a CP-based environment. Constraint programming specific features are used to ease the search process, still mantaining an absolute generality of the approach. We also propose a search strategy called Sliced Neighborhood Search, SNS, that iteratively explores slices of large neighborhoods of an incumbent solution by performing CP-based tree search and encloses concepts from metaheuristic techniques. SNS can be used as a stand alone search strategy, but it can alternatively be embedded in existing strategies as intensification and diversification mechanism. In particular we show its integration within the CP-based local branching. We provide an extensive experimental evaluation of the proposed approaches on instances of the Asymmetric Traveling Salesman Problem and of the Asymmetric Traveling Salesman Problem with Time Windows. The proposed approaches achieve good results on practical size problem, thus demonstrating the benefit of integrating metaheuristic concepts in CP-based frameworks.

Problem-specific design of metaheuristics for constrained spanning tree problems

When designing metaheuristic optimization methods, there is a trade-off between application range and effectiveness. For large real-world instances of combinatorial optimization problems out-of-the-box metaheuristics often fail, and optimization methods need to be adapted to the problem at hand. Knowledge about the structure of high-quality solutions can be exploited by introducing a so called bias into one of the components of the metaheuristic used. These problem-specific adaptations allow to increase search performance. This thesis analyzes the characteristics of high-quality solutions for three constrained spanning tree problems: the optimal communication spanning tree problem, the quadratic minimum spanning tree problem and the bounded diameter minimum spanning tree problem. Several relevant tree properties, that should be explored when analyzing a constrained spanning tree problem, are identified. Based on the gained insights on the structure of high-quality solutions, efficient and robust solution approaches are designed for each of the three problems. Experimental studies analyze the performance of the developed approaches compared to the current state-of-the-art.

Contributions to exact approaches in combinatorial optimization

The focus of this thesis is to contribute to the development of new, exact solution approaches to different combinatorial optimization problems. In particular, we derive dedicated algorithms for a special class of Traveling Tournament Problems (TTPs), the Dial-A-Ride Problem (DARP), and the Vehicle Routing Problem with Time Windows and Temporal Synchronized Pickup and Delivery (VRPTWTSPD). Furthermore, we extend the concept of using dual-optimal inequalities for stabilized Column Generation (CG) and detail its application to improved CG algorithms for the cutting stock problem, the bin packing problem, the vertex coloring problem, and the bin packing problem with conflicts. In all approaches, we make use of some knowledge about the structure of the problem at hand to individualize and enhance existing algorithms. Specifically, we utilize knowledge about the input data (TTP), problem-specific constraints (DARP and VRPTWTSPD), and the dual solution space (stabilized CG). Extensive computational results proving the usefulness of the proposed methods are reported.

Climbing Combinatorial Fitness Landscapes

International audience

Exact Combinatorial Optimization with Graph Convolutional Neural Networks

Combinatorial optimization problems are typically tackled by the branch-and-bound paradigm. We propose to learn a variable selection policy for branch-and-bound in mixed-integer linear programming, by imitation learning on a diversified variant of the strong branching expert rule. We encode states as bipartite graphs and parameterize the policy as a graph convolutional neural network. Experiments on a series of synthetic problems demonstrate that our approach produces policies that can improve upon expert-designed branching rules on large problems, and generalize to instances significantly larger than seen during training.

Empirical evidence for ultrametric structure in multi-layer perceptron error surfaces

Combinatorial optimization problems share an interesting property with spin glass systems in that their state spaces can exhibit ultrametric structure. We use sampling methods to analyse the error surfaces of feedforward multi-layer perceptron neural networks learning encoder problems. The third order statistics of these points of attraction are examined and found to be arranged in a highly ultrametric way. This is a unique result for a finite, continuous parameter space. The implications of this result are discussed.

MASDSheGATS – Scheduling System for Dynamic Manufacturing Environments

This chapter addresses the resolution of scheduling in manufacturing systems subject to perturbations. The planning of Manufacturing Systems involves frequently the resolution of a huge amount and variety of combinatorial optimisation problems with an important impact on the performance of manufacturing organisations. Examples of those problems are the sequencing and scheduling problems in manufacturing management, routing and transportation, layout design and timetabling problems.

Ant colony search algorithm for the optimal power flow problem

To maintain a power system within operation limits, a level ahead planning it is necessary to apply competitive techniques to solve the optimal power flow (OPF). OPF is a non-linear and a large combinatorial problem. The Ant Colony Search (ACS) optimization algorithm is inspired by the organized natural movement of real ants and has been successfully applied to different large combinatorial optimization problems. This paper presents an implementation of Ant Colony optimization to solve the OPF in an economic dispatch context. The proposed methodology has been developed to be used for maintenance and repairing planning with 48 to 24 hours antecipation. The main advantage of this method is its low execution time that allows the use of OPF when a large set of scenarios has to be analyzed. The paper includes a case study using the IEEE 30 bus network. The results are compared with other well-known methodologies presented in the literature.

Aspectos de aprendizagem em optimização

A optimização e a aprendizagem em Sistemas Multi-Agente são consideradas duas áreas promissoras mas relativamente pouco exploradas. A optimização nestes ambientes deve ser capaz de lidar com o dinamismo. Os agentes podem alterar o seu comportamento baseando-se em aprendizagem recente ou em objectivos de optimização. As estratégias de aprendizagem podem melhorar o desempenho do sistema, dotando os agentes da capacidade de aprender, por exemplo, qual a técnica de optimização é mais adequada para a resolução de uma classe particular de problemas, ou qual a parametrização é mais adequada em determinado cenário. Nesta dissertação são estudadas algumas técnicas de resolução de problemas de Optimização Combinatória, sobretudo as Meta-heurísticas, e é efectuada uma revisão do estado da arte de Aprendizagem em Sistemas Multi-Agente. É também proposto um módulo de aprendizagem para a resolução de novos problemas de escalonamento, com base em experiência anterior. O módulo de Auto-Optimização desenvolvido, inspirado na Computação Autónoma, permite ao sistema a selecção automática da Meta-heurística a usar no processo de optimização, assim como a respectiva parametrização. Para tal, recorreu-se à utilização de Raciocínio baseado em Casos de modo que o sistema resultante seja capaz de aprender com a experiência adquirida na resolução de problemas similares. Dos resultados obtidos é possível concluir da vantagem da sua utilização e respectiva capacidade de adaptação a novos e eventuais cenários.

A public transport bus assignment problem: parallel metaheuristics assessment

Combinatorial Optimization Problems occur in a wide variety of contexts and generally are NP-hard problems. At a corporate level solving this problems is of great importance since they contribute to the optimization of operational costs. In this thesis we propose to solve the Public Transport Bus Assignment problem considering an heterogeneous fleet and line exchanges, a variant of the Multi-Depot Vehicle Scheduling Problem in which additional constraints are enforced to model a real life scenario. The number of constraints involved and the large number of variables makes impracticable solving to optimality using complete search techniques. Therefore, we explore metaheuristics, that sacrifice optimality to produce solutions in feasible time. More concretely, we focus on the development of algorithms based on a sophisticated metaheuristic, Ant-Colony Optimization (ACO), which is based on a stochastic learning mechanism. For complex problems with a considerable number of constraints, sophisticated metaheuristics may fail to produce quality solutions in a reasonable amount of time. Thus, we developed parallel shared-memory (SM) synchronous ACO algorithms, however, synchronism originates the straggler problem. Therefore, we proposed three SM asynchronous algorithms that break the original algorithm semantics and differ on the degree of concurrency allowed while manipulating the learned information. Our results show that our sequential ACO algorithms produced better solutions than a Restarts metaheuristic, the ACO algorithms were able to learn and better solutions were achieved by increasing the amount of cooperation (number of search agents). Regarding parallel algorithms, our asynchronous ACO algorithms outperformed synchronous ones in terms of speedup and solution quality, achieving speedups of 17.6x. The cooperation scheme imposed by asynchronism also achieved a better learning rate than the original one.

A multi-objective model for a multi-period distribution management problem

The problems arising in commercial distribution are complex and involve several players and decision levels. One important decision is relatedwith the design of the routes to distribute the products, in an efficient and inexpensive way.This article deals with a complex vehicle routing problem that can beseen as a new extension of the basic vehicle routing problem. The proposed model is a multi-objective combinatorial optimization problemthat considers three objectives and multiple periods, which models in a closer way the real distribution problems. The first objective is costminimization, the second is balancing work levels and the third is amarketing objective. An application of the model on a small example, with5 clients and 3 days, is presented. The results of the model show the complexity of solving multi-objective combinatorial optimization problems and the contradiction between the several distribution management objective.

Adaptive approach heuristics for the generalized assignment problem

The Generalized Assignment Problem consists in assigning a setof tasks to a set of agents with minimum cost. Each agent hasa limited amount of a single resource and each task must beassigned to one and only one agent, requiring a certain amountof the resource of the agent. We present new metaheuristics forthe generalized assignment problem based on hybrid approaches.One metaheuristic is a MAX-MIN Ant System (MMAS), an improvedversion of the Ant System, which was recently proposed byStutzle and Hoos to combinatorial optimization problems, and itcan be seen has an adaptive sampling algorithm that takes inconsideration the experience gathered in earlier iterations ofthe algorithm. Moreover, the latter heuristic is combined withlocal search and tabu search heuristics to improve the search.A greedy randomized adaptive search heuristic (GRASP) is alsoproposed. Several neighborhoods are studied, including one basedon ejection chains that produces good moves withoutincreasing the computational effort. We present computationalresults of the comparative performance, followed by concludingremarks and ideas on future research in generalized assignmentrelated problems.