Improved methods for solving traffic flow problems in dynamic networks

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2002.Includes bibliographical references (p. 107-109).by Nathaniel J. Grier.S.M

Наблюдатель изменений в лесах кратчайших путей на динамических графах транспортных сетей

The purpose of the work is the development of basic data structures, speed-efficient and memoryefficient algorithms for tracking changes in predefined decisions about sets of shortest paths on transport networks, notifications about which are received by autonomous coordinated transport agents with centralized or collective control. A characteristic feature of transport operations is the independence and asynchrony of the emergence of perturbations of optimal solutions, as well as the lack of global influence of individual perturbations on the set of all processes on the network. This clearly determines the feasibility of realizing the idea of reoptimizing existing solutions in real time as information is received about disturbances in the structure and parameters of the transport network, various restrictions on the use of existing shortest paths. In contrast to the classical problems of finding shortest paths on static or dynamic graphs, it is proposed to supplement the set of situations controlled by the observer by taking into account the associations of shortest path trees with agents that actually use such paths. This will improve the responsiveness of agent notification processes for timely switching to a new path. The space of search states is a dynamically generated bipartite sparse graph of the transport network, represented by a list of arcs. The basic algorithm for finding the shortest paths uses Dijkstra's scheme, but implements a bootstrapping method to generate the search result. The compactness of the representation of the observed forest of shortest paths is achieved by mapping individual trees of such a forest onto the projection of tree vertices in memory, where the position of each vertex corresponds to the distance from the tree root. The proposed version of the construction of the search procedure is based on the mechanisms existing in database management systems for creating different relational representations of the physical data model. This eliminates the need to solve technological problems of complexing heterogeneous models of dynamic transport networks, memory allocation. As a result, the specification of various rules for the logistics of transport operations is simplified, since such operations in terms of object-oriented models are easily determined by polymorphic classes of transitions between nodes of the transport network.Цель работы – разработка базовых структур данных и эффективных по быстродействию и памяти алгоритмов слежения за изменениями предопределенных решений о множествах кратчайших путей на транспортных сетях, уведомления о которых получают автономные координируемые транспортные агенты с централизованным или коллективным управлением. Характерная особенность транспортных операций – независимость и асинхронность появления возмущений оптимальных решений, а также отсутствие глобального влияния отдельных возмущений на множество всех процессов на сети. Тем самым, очевидно, определяется целесообразность реализации идеи реоптимизации существующих решений в реальном времени по мере поступления информации о возмущениях структуры и параметров транспортной сети, различных ограничений на использование существующих кратчайших путей. В отличие от классических задач поиска кратчайших путей на статических или динамических графах, набор контролируемых наблюдателем ситуаций предлагается дополнить учетом ассоциаций деревьев кратчайших путей с фактически использующими такие пути агентами. Это позволит улучшить реактивность процессов уведомления агентов для своевременного переключения на новый путь. Пространство состояний поиска – динамически формируемый двудольный разреженный граф транспортной сети, представленный списком дуг. Базовый алгоритм поиска кратчайших путей использует схему Дейкстры, но реализует для формирования результата поиска метод бутстрэппинга. Компактность представления наблюдаемого леса кратчайших путей достигается отображением отдельных деревьев такого леса на проекцию вершин дерева в памяти, где позиция каждой вершины соответствует расстоянию от корня дерева. Предложенная версия построения процедуры поиска опирается на существующие в системах управления базами данных механизмы создания разных реляционных представлений физической модели данных. Это избавляет от необходимости решения технологических задач комплексирования гетерогенных моделей динамических транспортных сетей, распределения памяти. В результате спецификация различных правил логистики транспортных операций упрощается, так как такие операции в терминах объектно-ориентированных моделей легко определяются полиморфными классами переходов между узлами транспортной сети

Optimizing and Reoptimizing: tackling static and dynamic combinatorial problems

As suggested by the title, in this thesis both static and dynamic problems of Operations Research will be addressed by either designing new procedures or adapting well-known algorithmic schemes. Specifically, the first part of the thesis is devoted to the discussion of three variants of the widely studied Shortest Path Problem, one of which is defined on dynamic graphs. Namely, first the Reoptimization of Shortest Paths in case of multiple and generic cost changes is dealt with an exact algorithm whose performance is compared with Dijkstra's label setting procedure in order to detect which approach has to be preferred. Secondly, the k-Color Shortest Path Problem is tackled. It is a recent problem, defined on an edge-constrained graph, for which a Dynamic Programming algorithm is proposed here; its performance is compared with the state of the art solution approach, namely a Branch & Bound procedure. Finally, the Resource Constrained Clustered Shortest Path Tree Problem is presented. It is a newly defined problem for which both a mathematical model and a Branch & Price procedure are detailed here. Moreover, the performance of this solution approach is compared with that of CPLEX solver. Furthermore, in the first part of the thesis, also the Path Planning in Urban Air Mobility, is discussed by considering both the definition of the Free-Space Maps and the computation of the trajectories. For the former purpose, three different but correlated discretization methods are described; as for the latter, a two steps resolution, offline and online, of the resulting shortest path problems is performed. In addition, it is checked whether the reoptimization algorithm can be used in the online step. In the second part of this thesis, the recently studied Additive Manufacturing Machine Scheduling Problem with not identical machines is presented. Specifically, a Reinforcement Learning Iterated Local Search meta-heuristic featuring a Q-learning Variable Neighbourhood Search is described to solve this problem and its performance is compared with the one of CPLEX solver. It is worthwhile mentioning that, for each of the proposed approaches, a thorough experimentation is performed and each Chapter is equipped with a detailed analysis of the results in order to appraise the performance of the method and to detect its limits

Comparison of heuristic approaches for the multiple depot vehicle scheduling problem

Given a set of timetabled tasks, the multi-depot vehicle scheduling problemis a well-known problem that consists of determining least-cost schedulesfor vehicles assigned to several depots such that each task is accomplishedexactly once by a vehicle. In this paper, we propose to compare theperformance of five different heuristic approaches for this problem,namely, a heuristic \\mip solver, a Lagrangian heuristic, a columngeneration heuristic, a large neighborhood search heuristic using columngeneration for neighborhood evaluation, and a tabu search heuristic. Thefirst three methods are adaptations of existing methods, while the last twoare novel approaches for this problem. Computational results on randomlygenerated instances show that the column generation heuristic performs thebest when enough computational time is available and stability is required,while the large neighborhood search method is the best alternative whenlooking for a compromise between computational time and solution quality.tabu search;column generation;vehicle scheduling;heuristics;Lagrangian heuristic;large neighborhood search;multiple depot

Minimum Spanning Trees in Weakly Dynamic Graphs

International audienceIn this paper, we study weakly dynamic undirected graphs, that can be used to represent some logistic networks. The goal is to deliver all the delivery points in the network. The network exists in a mostly stable environment, except for a few edges known to be non-stable. The weight of each of these non-stable edges may change at any time (bascule or lift bridge, elevator, traffic congestion...). All other edges have stable weights that never change. This problem can be now considered as a Minimum Spanning Tree (MST) problem on a dynamic graph. We propose an efficient polynomial algorithm that computes in advance alternative MSTs for all possible configurations. No additional computation is then needed after any change in the problem because the MSTs are already known in all cases. We use these results to compute critical values for the non-stable weights and to pre-compute best paths. When the non-stable weights change, the appropriate MST may then directly and immediately be used without any recomputation

The dynamic, resource-constrained shortest path problem on an acyclic graph with application in column generation and literature review on sequence-dependent scheduling

This dissertation discusses two independent topics: a resource-constrained shortest-path problem (RCSP) and a literature review on scheduling problems involving sequence-dependent setup (SDS) times (costs). RCSP is often used as a subproblem in column generation because it can be used to solve many practical problems. This dissertation studies RCSP with multiple resource constraints on an acyclic graph, because many applications involve this configuration, especially in column genetation formulations. In particular, this research focuses on a dynamic RCSP since, as a subproblem in column generation, objective function coefficients are updated using new values of dual variables at each iteration. This dissertation proposes a pseudo-polynomial solution method for solving the dynamic RCSP by exploiting the special structure of an acyclic graph with the goal of effectively reoptimizing RCSP in the context of column generation. This method uses a one-time âÂÂpreliminaryâ phase to transform RCSP into an unconstrained shortest path problem (SPP) and then solves the resulting SPP after new values of dual variables are used to update objective function coefficients (i.e., reduced costs) at each iteration. Network reduction techniques are considered to remove some nodes and/or arcs permanently in the preliminary phase. Specified techniques are explored to reoptimize when only several coefficients change and for dealing with forbidden and prescribed arcs in the context of a column generation/branch-and-bound approach. As a benchmark method, a label-setting algorithm is also proposed. Computational tests are designed to show the effectiveness of the proposed algorithms and procedures. This dissertation also gives a literature review related to the class of scheduling problems that involve SDS times (costs), an important consideration in many practical applications. It focuses on papers published within the last decade, addressing a variety of machine configurations - single machine, parallel machine, flow shop, and job shop - reviewing both optimizing and heuristic solution methods in each category. Since lot-sizing is so intimately related to scheduling, this dissertation reviews work that integrates these issues in relationship to each configuration. This dissertation provides a perspective of this line of research, gives conclusions, and discusses fertile research opportunities posed by this class of scheduling problems. since, as a subproblem in column generation, objective function coefficients are updated using new values of dual variables at each iteration. This dissertation proposes a pseudo-polynomial solution method for solving the dynamic RCSP by exploiting the special structure of an acyclic graph with the goal of effectively reoptimizing RCSP in the context of column generation. This method uses a one-tim

Improved Algorithms for the Steiner Problem in Networks

We present several new techniques for dealing with the Steiner problem in (undirected) networks. We consider them as building blocks of an exact algorithm, but each of them could also be of interest in its own right. First, we consider some relaxations of integer programming formulations of this problem and investigate different methods for dealing with these relaxations, not only to obtain lower bounds, but also to get additional information which is used in the computation of upper bounds and in reduction techniques. Then, we modify some known reduction tests and introduce some new ones. We integrate some of these tests into a package with a small worst case time which achieves impressive reductions on a wide range of instances. On the side of upper bounds, we introduce the new concept of heuristic reductions. On the basis of this concept, we develop heuristics that achieve sharper upper bounds than the strongest known heuristics for this problem despite running times which are smaller by orders of magnitude. Finally, we integrate these blocks into an exact algorithm. We present computational results on a variety of benchmark instances. The results are clearly superior to those of all other exact algorithms known to the authors

The Quick Time Dependent Quickest Flow Problem: A Lesson in Zero-Sum Cycles

A quick solution technique for the integral time-dependent quickest flow problem with no waiting is presented. The proposed technique is based on the successive shortest path approach and modifies an existing algorithm to improve its average performance. At each iteration, a reoptimization procedure is employed to determine the augmenting path given updates to the residual graph. The residual graph, by construction, almost always contains zero-sum cycles when employed in this context. These zero-sum cycles pose a unique problem for the reoptimization technique. A heuristic that can be embedded in the reoptimization algorithm to provide path solutions in the presence of zero-sum cycles has been proposed. In the computational experiments, the heuristic provided an optimal solution nearly 100% of the times. Further, a modified implementation of an existing path-finding algorithm has been used to solve the time-dependent quickest flow problem with source waiting