Dynamic Network Flow Research Articles

Mixed-integer programming models and heuristic algorithms for the maximum value dynamic network flow scheduling problem

Various applications in contested logistics and infrastructure restoration require dynamic flow solutions characterized by a schedule of network flows consecutively transmitted over a sequence of successive periods. For these schedules, we assume flows transmit via arcs during periods while flows reside at nodes from one period to the next. Within this context, we introduce the Maximum Value Dynamic Network Flow Problem (MVDFP) in which we seek to maximize the cumulative value of a non-simultaneous network flow schedule that accumulates node value whenever some minimum amount of flow resides at a node between periods. For solving the MVDFP, we first introduce a large mixed-integer program (MIP). As this MIP can become computationally-expensive for large networks, we present a trio of computationally-effective, easy to implement heuristic approaches that solve a series of smaller, more manageable MIPs. These heuristic approaches typically determine high-quality solutions significantly faster than the MIP obtains an optimal solution by dividing the full network flow schedule into a sequence of consecutive shorter network flow subschedules. In many cases, at least one of our heuristic approaches produces an optimal solution in a fraction of the MIP’s computational time. We present extensive computational results to highlight our heuristics’ efficacy, discuss for what instances each approach may be most applicable, and detail future research avenues.

Modeling the Great Lakes Basin based on dynamic network flows

Modeling the Great Lakes Basin based on dynamic network flows

Sink location problems in dynamic flow grid networks

A dynamic flow network consists of a directed graph, where nodes called sources represent locations of evacuees, and nodes called sinks represent locations of evacuation facilities. Each source and each sink are given supply representing the number of evacuees and demand representing the maximum number of acceptable evacuees, respectively. Each edge is given capacity and transit time. Here, the capacity of an edge bounds the rate at which evacuees can enter the edge per unit time, and the transit time represents the time which evacuees take to travel across the edge. The evacuation completion time is the minimum time at which each evacuee can arrive at one of the evacuation facilities. Given a dynamic flow network without sinks, once sinks are located on some nodes or edges, the evacuation completion time for this sink location is determined. We then consider the problem of locating sinks to minimize the evacuation completion time, called the sink location problem. The problems have been given polynomial-time algorithms only for limited networks such as paths [1–3], cycles [1], and trees [4–6], but no polynomial-time algorithms are known for more complex network classes. In this paper, we prove that the 1-sink location problem can be solved in polynomial-time when an input network is a grid with uniform edge capacity and transit time.

Dynamics of a susceptible-infected-recovered model on complex networks with interregional migration.

We present a susceptible-infected-recovered model based on a dynamic flow network that describes the epidemic process on complex metapopulation networks. This model views population regions as interconnected nodes and describes the evolution of each region using a system of differential equations. The next-generation matrix method is used to derive the global basic reproduction number for three cases: a general network with homogeneous infection rates in all regions, a fully connected network, and a star network with heterogeneous infection and recovery rates. For the homogeneous case, we show that this global basic reproduction number is independent of the migration rates between regions. However, the rate of convergence of each region to an equilibrium state exhibits a much larger variance in random (Erdős-Rényi) networks compared to small-scale (Barabási-Albert) networks. For the general heterogeneous case, we report interesting results, namely that the global basic reproduction number decays exponentially with respect to the smallest nonzero Laplacian eigenvalue (algebraic connectivity). Furthermore, we demonstrate both analytically and numerically that as the network's algebraic connectivity increases, either by increasing the average node degree of each region or the global migration rate, the global basic reproduction number decreases and converges to the ratio of the average local infection rate to the average local recovery rate, meaning that the lower bound of the global basic reproduction rate does not equal the mean of local basic reproduction rates.

Resilient Control of Dynamic Flow Networks Subject to Stochastic Cyber-Physical Disruptions

Resilient Control of Dynamic Flow Networks Subject to Stochastic Cyber-Physical Disruptions

A Hybrid Control Path Planning Architecture Based on Traffic Equilibrium Assignment for Emergency

Unconventional events exacerbate the imbalance between regional transportation demand and limited road network resources. Scientific and efficient path planning serves as the foundation for rapidly restoring equilibrium to the road network. In real large-scale road networks, especially during emergencies, it is usually difficult to obtain or predict accurate dynamic traffic network flows in real-time, which is used to support equilibrium path planning. Moreover, the traditional iterative methods cannot meet the real-time demand of emergency equilibrium path planning decision generation. To solve the above problems, this paper proposes a hybrid control architecture for path planning based on equilibrium traffic assignment theory. The architecture introduces the travelers’ real-time travel data and constructs a spatio-temporal neural network, which captures the evolution of traffic network loads. Adaptive multi-graph fusion technology is used to mix the background traffic flow data and the traveler’s real-time Origin–Destination (OD) data, to mine the dynamic correlation between the traffic state and the travelers’ travel. Based on the real-time prediction of dynamic network states, equilibrium mapping learning is carried out to pre-allocate potential travel demands and construct equilibrium traffic graphs based on system optimization traffic assignment. Finally, individual evacuation path strategies are generated online in a data-driven manner in real time to achieve improved resilience in the transportation system.

The Strong Integral Input-to-State Stability Property in Dynamical Flow Networks

Dynamical flow networks serve as macroscopic models for, e.g., transportation networks, queuing networks, and distribution networks. While the flow dynamics in such networks follow the conservation of mass on the links, the outflow from each link is often non-linear due to, e.g., flow capacity constraints and simultaneous service rate constraints. Such non-linear constraints imply a limit on the magnitude of exogenous inflow that is able to be accommodated by the network before it becomes overloaded and its state trajectory diverges. This paper shows how the Strong integral Input-to-State Stability (Strong iISS) property allows for quantifying the effects of the exogenous inflow on the flow dynamics. The Strong iISS property enables a unified stability analysis of classes of dynamical flow networks that were only partly analyzable before, such as networks with cycles, multi-commodity flow networks and networks with non-monotone flow dynamics. We present sufficient conditions on the maximum magnitude of exogenous inflow to guarantee input-to-state stability for a dynamical flow network, and we also present cases when this sufficient condition is necessary. The conditions are exemplified on a few existing dynamical flow network models, specifically, fluid queuing models with time-varying exogenous inflows and multi-commodity flow models.

Optimizing oil spill emergency logistics: a time-varying multi-resource collaborative scheduling model.

Emergency resource scheduling is at the heart of the response to an oil spill, as it lays the foundation for all other emergency operations. Extant studies address the dynamicity inherent to these operations primarily by modeling a dynamic network flow with static data, which is not applicable to continuously changing conditions resulting from oil film movement. To enhance the responsiveness and cost-efficiency of the response to oil spills, this paper takes a novel approach and formulates a multi-objective location-routing model for multi-resource collaborative scheduling, namely, harnessing time-varying parameters rather than static data to model real-time changes in the demand for emergency resources and the transportation network. Additionally, the model considers various operational factors, including the transportation of multiple resources in the order of operating procedures; the coordination of split delivery with the consumption of emergency resources; and the matching of multiple resources with suitable vehicles. To solve the proposed model, a hybrid heuristic algorithm of PSO-PGSA is developed, which utilizes particle swarm optimization (PSO) to search widely for non-dominated solutions. The algorithm then makes use of the plant growth simulation algorithm (PGSA) to find the more effective vehicle routes based on the obtained solutions. Finally, a numerical analysis is used to illustrate the practical capabilities of the developed model and solution strategies. Most significantly, our work not only validates the methodology proposed here but also underlines the importance of incorporating the features of an oil spill emergency response into emergency logistics in general.

Scalable Computation of Dynamic Flow Problems via Multimarginal Graph-Structured Optimal Transport

In this work, we develop a new framework for dynamic network flow problems based on optimal transport theory. We show that the dynamic multicommodity minimum-cost network flow problem can be formulated as a multimarginal optimal transport problem, where the cost function and the constraints on the marginals are associated with a graph structure. By exploiting these structures and building on recent advances in optimal transport theory, we develop an efficient method for such entropy-regularized optimal transport problems. In particular, the graph structure is utilized to efficiently compute the projections needed in the corresponding Sinkhorn iterations, and we arrive at a scheme that is both highly computationally efficient and easy to implement. To illustrate the performance of our algorithm, we compare it with a state-of-the-art linear programming (LP) solver. We achieve good approximations to the solution at least one order of magnitude faster than the LP solver. Finally, we showcase the methodology on a traffic routing problem with a large number of commodities. Funding: This work was supported by KTH Digital Futures, Knut och Alice Wallenbergs Stiftelse [Grants KAW 2018.0349, KAW 2021.0274, the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation], Vetenskapsrådet [Grant 2020-03454], and the National Science Foundation [Grants 1942523 and 2206576].

Scalable Computation of Dynamic Flow Problems via Multimarginal Graph-Structured Optimal Transport

In this work, we develop a new framework for dynamic network flow problems based on optimal transport theory. We show that the dynamic multicommodity minimum-cost network flow problem can be formulated as a multimarginal optimal transport problem, where the cost function and the constraints on the marginals are associated with a graph structure. By exploiting these structures and building on recent advances in optimal transport theory, we develop an efficient method for such entropy-regularized optimal transport problems. In particular, the graph structure is utilized to efficiently compute the projections needed in the corresponding Sinkhorn iterations, and we arrive at a scheme that is both highly computationally efficient and easy to implement. To illustrate the performance of our algorithm, we compare it with a state-of-the-art linear programming (LP) solver. We achieve good approximations to the solution at least one order of magnitude faster than the LP solver. Finally, we showcase the methodology on a traffic routing problem with a large number of commodities. Funding: This work was supported by KTH Digital Futures, Knut och Alice Wallenbergs Stiftelse [Grants KAW 2018.0349, KAW 2021.0274, the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation], Vetenskapsrådet [Grant 2020-03454], and the National Science Foundation [Grants 1942523 and 2206576].

A design method for edge–cloud collaborative product service system: a dynamic event-state knowledge graph-based approach with real case study

ABSTRACT Product service system (PSS) is an effective approach to achieve win–win situation between service providers and consumers, and the design of PSS is the first step of its application. However, on the one hand, PSS is relatively an abstract concept, and what exactly should be considered when designing a PSS still require further exploration; on the other hand, the fast development of information and web technologies bring both opportunities and challenges for PSS design. For example, how to efficiently design the smart & connected products that support remote monitoring and control of service operation, how to build the dynamic service activity flow model that can be intuitively and visually read by human engineers and also can be conveniently deployed on computers, and how to realise edge–cloud collaboration between service providers and consumers. In this regard, a service requirement-oriented four-step generic PSS design method is established, including service mode selection and structured service order generation → smart & connected service product configuration → dynamic event-state knowledge graph-based service activity flow and service resource network configuration → industrial internet-based edge–cloud collaborative service delivering. Finally, a real design case of carbon block grinding and polishing PSS is used for verification. ABBREVIATIONS: AS: assembly service; C: controlling targets or objectives; I: input; M: enabling methods or mechanisms; MRO: maintenance, repair, and operation; MS: maintenance service; O: output; OEE: overall equipment effectiveness; PSS: product service system; PV: processing volume; R x E y -S z : the relation from Event y to the State z of Service activity x; R x S z -E y : the relation from the State z of Service activity x to Event y; StateAi Current: the current state of Service activity i; StateAi Futurek : the kth possible future state of Service activity i; StateAi Historyj : the jth historical state of service activity i; StateEy : the state of Event y; Tq : the qth time point in the time line.

Collaborative optimal scheduling of integrated energy system considering carbon trading

In order to further realize the zero carbon transition of the energy system, this paper proposes a robust optimal scheduling model for transmission and distribution collaboration considering carbon trading. In this model, the multi energy equipment in the integrated energy system is modeled for transmission and distribution coordination. The long-distance power transmission network is an electricity gas coupling network, and the energy hub is the park distribution network. The nonconvex dynamic power flow network is relaxed by using the method of second-order cone programming, and the ADMM method is used to ensure that the optimal energy coordination is obtained between regions under the limited information exchange, so as to ensure user privacy. The validity of the scheduling model is proved by the case study.

Modeling Minimum Cost Network Flows With Port‐Hamiltonian Systems

AbstractWe give a short overview of advantages and drawbacks of the classical formulation of minimum cost network flow problems and solution techniques, to motivate a reformulation of classical static minimum cost network flow problems as optimal control problems constrained by port‐Hamiltonian systems (pHS). The first‐order optimality system for the port‐Hamiltonian system‐constrained optimal control problem is formally derived. Then we propose a gradient‐based algorithm to find optimal controls. The port‐Hamiltonian system formulation naturally conserves flow and supports a wide array of further modeling options as, for example, node reservoirs, flow dependent costs, leaking pipes (dissipation) and coupled sub‐networks (ports). They thus provide a versatile alternative to state‐of‐the art approaches towards dynamic network flow problems, which are often based on computationally costly time‐expanded networks. We argue that this opens the door for a plethora of modeling options and solution approaches for network flow problems.

SOME NETWORKS THAT ALLOW SPLASHES OF DYNAMIC FLOWS AND FINDING THE MAXIMUM SPLASH VALUE

This paper is devoted to dynamic network flows. A phenomenon called dynamic network flow splash is studied (splash is an exceeding of flow value over maximum feasible static flow for the same network). Parallel and tree-structured network topology classes are defined. Conditions for occurring network flow splash in the parallel and tree-structured networks are discovered. A formula for computing a value of maximum dynamic network flow splash is obtained. An algorithm for calculating the formula values is presented.

Optimized Compiler for Distributed Quantum Computing

Practical distributed quantum computing requires the development of efficient compilers, able to make quantum circuits compatible with some given hardware constraints. This problem is known to be tough, even for local computing. Here, we address it on distributed architectures. As generally assumed in this scenario, telegates represent the fundamental remote (inter-processor) operations. Each telegate consists of several tasks: (i) entanglement generation and distribution, (ii) local operations, and (iii) classical communications. Entanglement generations and distribution is an expensive resource, as it is time-consuming. To mitigate its impact, we model an optimization problem that combines running-time minimization with the usage of distributed entangled states. Specifically, we formulated the distributed compilation problem as a dynamic network flow. To enhance the solution space, we extend the formulation, by introducing a predicate that manipulates the circuit given in input and parallelizes telegate tasks. To evaluate our framework, we split the problem into three sub-problems, and solve it by means of an approximation routine. Experiments demonstrate that the run-time is resistant to the problem size scaling. Moreover, we apply the proposed algorithm to compile circuits under different topologies, showing that topologies with a higher ratio between edges and nodes give rise to shallower circuits.

Minmax Centered k-Partitioning of Trees and Applications to Sink Evacuation with Dynamic Confluent Flows

Let $T=(V,E)$ be a tree with associated costs on its subtrees. A minmax $k$-partition of $T$ is a partition into $k$ subtrees, minimizing the maximum cost of a subtree over all possible partitions. In the centered version of the problem, the cost of a subtree cost is defined as the minimum cost of "servicing" that subtree using a center located within it. The problem motivating this work was the sink-evacuation problem on trees, i.e., finding a collection of $k$-sinks that minimize the time required by a confluent dynamic network flow to evacuate all supplies to sinks. This paper provides the first polynomial-time algorithm for solving this problem, running in $O\Bigl(\max(k,\log n) k n \log^4 n\Bigr)$ time. The technique developed can be used to solve any Minmax Centered $k$-Partitioning problem on trees in which the servicing costs satisfy some very general conditions. Solutions can be found for both the discrete case, in which centers must be on vertices, and the continuous case, in which centers may also be placed on edges. The technique developed also improves previous results for finding a minmax cost $k$-partition of a tree given the location of the sinks in advance.

Node Centrality Comparison between Bus Line and Passenger Flow Networks in Beijing

In recent decades, complex network theory has become one of the most important approaches for exploring the structure and dynamics of traffic networks. Most studies mainly focus on the static topology features of the traffic networks, and there are also increasing literature focusing on passenger flow networks. However, not much work has been completed on comparing the static networks with dynamic flow networks from the perspective of supply and demand. Therefore, this study aimed to apply the complex network approach to explore the spatial relationship between bus line organization and bus flows in Beijing. Based on the bus route data and the passenger flow data obtained from the Beijing smart bus card, this study investigated the spatial characteristics of the bus line network and the temporal bus flow networks, and presented a comparison analysis on the spatial relationship between them by using the node centrality indices, namely degree centrality, betweenness centrality and closeness centrality. The results show that the overall spatial patterns of node centralities between the bus line network and the bus flow network were similar, while there were also some differences. For weekdays, the correlation between them is higher, as calculated by the degree of centrality. For weekends, the two networks have a greater correlation measured by degree centrality and betweenness centrality. The highest coefficients of correlation between the line network and traffic network appear in the morning peak, which implies that the congestion issues during the morning peak hours might receive the highest priority in Beijing’s bus-line network planning. Our study can provide implications for policymakers to improve the public urban transport network, and thus enhance residents’ happiness.

Multiclass dynamic emergency traffic collaborative optimization considering multiple solutions with stage-based algorithm

Multiclass dynamic emergency traffic consisting of evacuation and rescue vehicles is a common situation on the road network in the response of the sudden local disaster, e.g., fire disaster, hazardous chemical disaster, etc. However, there are several options for the optimal rescue traffic routes sharing the same optimum objective value during the evacuation, and it is not clear how different optimal rescue traffic routes will affect evacuation traffic. Our motivation is based on the possibility for the dynamic rescue traffic optimum to have multiple solutions on the road network shared by multiple vehicle classes, which differ in when and which roads are occupied by how many rescue vehicles during the evacuation. To this end, we formulate multiclass dynamic emergency traffic collaborative optimization problem considering multiple solutions and traffic priority as the MMILP-RRO formulation, which consists of multi-objective mixed integer linear programming formulation (MMILP) and risk-based rescue traffic optimization formulation (RRO), and is solved by a stage-based optimization algorithm. The numerical results are conducted on the Beijing’s Zhongguancun​ road network to provide some insights into the effects of multiple optimal rescue traffic routes on evacuation traffic for multiclass dynamic emergency traffic network flow collaborative management.

Robust building evacuation planning in a dynamic network flow model under collapsible nodes and arcs

Rapid urbanization has caused various social problems. One typical example is the high population density of a building, particularly in a commercial building or a mega-mall. When an emergency, such as a natural or human-made disaster, occurs in a building with a high population, establishing a proper evacuation plan is required to minimize casualties. Accordingly, the evacuation planning problem, which determines optimal routes for evacuees from disaster-prone areas to safe areas, has been actively studied in various fields. However, research considering the possibility of further collapse of a specific area or intermediate route in the building has been overlooked. We propose a robust evacuation planning problem based on a dynamic network flow model that determines the optimal routes for evacuees from a building that has the potential to collapse. Computational results show that routes passing through areas with the potential to collapse may or may not be optimal for evacuees, depending on the given timeframe. If the timeframe is sufficient, detouring around the collapsible areas could be the optimal plan; however, if the timeframe is insufficient, passing through collapsible areas, with taking the risk, could be the optimal plan.

Dynamic Network Flow Model for Power Grid Systemic Risk Assessment and Resilience Enhancement

Dynamic Network Flow Model for Power Grid Systemic Risk Assessment and Resilience Enhancement