Flows over time with load-dependent transit times

Flow variation over time is an important feature in network flow problems arising in various applications such as road or air traffic control, production systems, communication networks (e.g., the Internet), and financial flows. Another crucial phenomenon in many of those applications is that the time taken to traverse an edge varies with the current amount of flow on this edge. Since it is already a highly nontrivial problem to map these two aspects into an appropriate and tractable mathematical network flow model, there are hardly any algorithmic techniques known which are capable of providing reasonable solutions even for networks of rather modest size

ارائه یک الگوريتم نموداری براي يافتن سريع¬ترين مسيرهاي امداد و نجات در شبکه ترافيک شهري

مقدمه: به‌طور کلی حمل و نقل روان، مؤثر و ايمن، يکي از زيرساخت¬هاي لازم براي توسعه صنايع، افزايش سطح رفاه و ارایه خدمات امدادی در هر کشور است. در شبکه¬های شلوغ ترافیکی یافتن بهترین مسیرها برای ارایه خدمات امدادی و اعزام اورژانس اهمیت ویژه¬ای دارد. در سال¬هاي اخير با توسعه سريع سيستم¬هاي هوشمند حمل و نقل، علاقه-مندی زیادی در زمينه مدل¬سازي و تعيين مسيرهای بهينه برای ارایه خدمات امداد و نجات در شبکه¬هاي حمل و نقلي که داراي رفتار پويا و جريان متغير با زمان هستند، بوجود آمده است. روش¬ها: در اين مقاله پس از انجام مطالعات کتابخانه‌ای، یک الگوريتم نموداری براي يافتن کوتاه-ترين فاصله زماني بين هر دو نقطه مفروض در يک شبکه ترافيک شهری و نيز تعيين مسير متناظر با کوتاه¬ترين فاصله زماني بين هر زوج از نقاط مذکور ارائه شده است. اين الگوريتم از تلفيق الگوريتم برنامه¬ريزي پوياي فلويد- وارشال به منظور يافتن کوتاه¬ترين فاصله زماني، الگوريتم پاي ( ) براي تعيين مسير متناظر با کوتاه‌ترين فاصله زماني و تابع ديويدسون براي در نظر گرفتن زمان انتقال در جريان¬هاي متغير تدوين شده است. سپس با استفاده از مطالعات میدانی و جمع‌آوری اطلاعات ترافیکی الگوریتم مذکور بر روی شبکه ترافیکی پیاده‌سازی شده است. یافته¬ها: الگوريتم معرفي شده بر روي شبکه ترافيک بخش مرکزي شهر شاهرود اجرا شده و کوتاه¬ترين فاصله زماني و مسير متناظر با آن بين دو نقطه مفروض از شبکه ترافيکي شهر جهت گسیل سریع وسایل نقلیه امدادی تعيين شده است. نتیجه¬گیری: با کمک روشي که در اين مقاله ارائه شده، سريع¬ترين مسيرهاي دسترسي از يک نقطه خاص به نقطه¬اي ديگر براي گسیل اتومبیل¬های اورژانس و ديگر وسايل نقليه امدادی، مشخص شده است

Robust Flows over Time: Models and Complexity Results

We study dynamic network flows with uncertain input data under a robust optimization perspective. In the dynamic maximum flow problem, the goal is to maximize the flow reaching the sink within a given time horizon $T$, while flow requires a certain travel time to traverse an edge. In our setting, we account for uncertain travel times of flow. We investigate maximum flows over time under the assumption that at most $\Gamma$ travel times may be prolonged simultaneously due to delay. We develop and study a mathematical model for this problem. As the dynamic robust flow problem generalizes the static version, it is NP-hard to compute an optimal flow. However, our dynamic version is considerably more complex than the static version. We show that it is NP-hard to verify feasibility of a given candidate solution. Furthermore, we investigate temporally repeated flows and show that in contrast to the non-robust case (that is, without uncertainties) they no longer provide optimal solutions for the robust problem, but rather yield a worst case optimality gap of at least $T$. We finally show that the optimality gap is at most $O(\eta k \log T)$, where $\eta$ and $k$ are newly introduced instance characteristics and provide a matching lower bound instance with optimality gap $\Omega(\log T)$ and $\eta = k = 1$. The results obtained in this paper yield a first step towards understanding robust dynamic flow problems with uncertain travel times

Shortest Paths and Probabilities on Time-Dependent Graphs - Applications to Transport Networks

International audienceIn this paper, we focus on time-dependent graphs which seem to be a good way to model transport Networks. In the first part, we remind some notations and techniques related to time-dependent graphs. In the second one, we introduce new algorithms to take into account the notion of probability related to paths in order to guarantee travelling times with a certain accuracy. We also discuss different probabilistic models and show the links between them

Optimal control for port-Hamiltonian systems and a new perspective on dynamic network flow problems

We formulate open-loop optimal control problems for general port-Hamiltonian systems with possibly state-dependent system matrices and prove their well-posedness. The optimal controls are characterized by the first-order optimality system, which is the starting point for the derivation of an adjoint-based gradient descent algorithm. Moreover, we discuss the relationship of port-Hamiltonian dynamics and minimum cost network flow problems. Our analysis is underpinned by a proof of concept, where we apply the proposed algorithm to static minimum cost flow problems and dynamic minimum cost flow problems with a simple directed acyclic graph. The numerical results validate the approach

An annotated overview of dynamic network flows

The need for more realistic network models led to the development of the dynamic network flow theory. In dynamic flow models it takes time for the flow to pass an arc, the flow can be delayed at nodes, and the network parameters, e.g., the arc capacities, can change in time. Surprisingly perhaps, despite being closer to reality, dynamic flow models have been overshadowed by the classical, static model. This is largely due to the fact that while very efficient solution methods exist for static flow problems, dynamic flow problems have proved to be more difficult to solve. Our purpose with this overview is to compensate for this eclipse and introduce dynamic flows to the interested reader. To this end, we present the main flow problems that can appear in a dynamic network, and review the literature for existing results about them. Our approach is solution oriented, as opposed to dealing with modelling issues. We intend to provide a survey that can be a first step for readers wondering whether a given dynamic network flow problem has been solved or not. Besides restating the problems, we also describe the main proposed solution methods. An additional feature of this paper is an annotated list of the most important references about the subject

An FPTAS for Quickest Multicommodity Flows with Inflow-Dependent Transit Times

Given a network with capacities and transit times on the arcs, the quickest flow problem asks for a "flow over time" that satisfies given demands within minimal time. In the setting of flows over time, flow on arcs may vary over time and the transit time of an arc is the time it takes for flow to travel through this arc. In most real-world applications (such as, e.g., road traffic, communication networks, production systems, etc.), transit times are not fixed but depend on the current flow situation in the network. We consider the model where the transit time of an arc is given as a non-decreasing function of the rate of inflow into the arc. We prove that the quickest s-t-flow problem is NP-hard in this setting and give various approximation results, including a fully polynomial time approximation scheme (FPTAS) for the quickest multicommodity flow problem with bounded cos

Shortest Paths and Probabilities on Time-Dependent Graphs - Applications to Transport Networks

International audienceIn this paper, we focus on time-dependent graphs which seem to be a good way to model transport Networks. In the first part, we remind some notations and techniques related to time-dependent graphs. In the second one, we introduce new algorithms to take into account the notion of probability related to paths in order to guarantee travelling times with a certain accuracy. We also discuss different probabilistic models and show the links between them