Maximum Dynamic Flow Research Articles

A Survey on Models of Flows over Time with Flow Dependent Transit Times

Ford and Fulkerson introduce the maximum dynamic flow problems with fixed transit times on the arcs and developed the first well known algorithm that sends maximum flow from the source to the sink by augmenting along s-t paths and prove the maximum amount of flow is equal to the total capacity of the arcs in minimum cut. In flows over time with fixed transit time on the arc, the time it takes to traverse an arc does not depend on the current flow situation on the arc. But in real situation, the flow units travelling on the same arc at the same time do not necessarily experience the same pace, i.e., flow units are in general not entering and leaving an arc in the same order. In this paper, we discuss the time expanded graph for load-dependent and inflow[1]dependent transit times. We also discuss the discrete and continuous time flow, earliest arrival flow and quickest transhipment problem.

Prioritized Max-FlowLoc for Evacuation Planning

From different disasters such as earthquakes, flooding, landslides, hurricanes, blizzards, etc., thousands of people have lost their lives and millions of people have to be rescued each year. To reduce the losses from different disasters, appropriate emergency plannings are essential. The allocation of facility in an appropriate location of prioritized network are evenly used in such plannings. The prioritized maximum flow problem is to obtain the maximum flow in a network by taking a predetermined leaving pattern at the sources and/or reaching pattern at the sinks. But placing the facility on an arc, which is a very important task to supply the basic needs of the evacuees, reduces the arc capacity so that the maximum flow value in the network may be reduced. Here, we introduce the prioritized maximum static and dynamic flow location (FlowLoc) problems in a single source and multi-sink network. The mathematical models are given for both single and multi facility problems. The polynomial time algorithms are presented in single facility cases. After proving the $\mathcal{NP}$-completeness, the polynomial-time heuristics are presented for optimal or near optimal solutions in case of multi-facility cases.

Lexicographic maximum dynamic evacuation modeling with partial lane reversal based on hesitant fuzzy TOPSIS

Decision-making under evacuation environment is extremely complex due to the variety of aspects to be considered as well as the necessity of high-speed reaction. Therefore, the elaboration of the evacuation pattern, particularly, finding the priority order of shelters for evacuation, requires multiple experts to be involved in the decision-making process. Moreover, decision-makers cannot provide precise assessments of alternatives due to the uncertainty inherent in the network parameters, dynamic nature of transportation, complexity of the task. In this regard, the method of finding the priority order of terminals for the evacuation based on the fuzzy hesitant TOPSIS method is proposed. The method handles a modified ranking index to find the priority order based on assigned weights of separations. In this method, experts’ assessments are presented as linguistic terms, which are further converted to fuzzy triangular numbers. This leads to the hesitant fuzzy TOPSIS decision-making with completely unknown attribute weights.The existing state of the art of evacuation modeling lacks fuzzy statements of evacuation flow problems. Our work aims to contribute to the design of the algorithm for the lexicographic maximum flow finding in the evacuation dynamic network with fuzzy transit arc capacities and transit traversal time parameters with partial contraflow. The approach reverses only necessary arcs by reversing the traffic in vacant segments, which allows using these segments for movement towards the safe areas as well. In addition, the method based on linear combinations of spreads is proposed to handle fuzzy values, which does not lead to the blurring of a fuzzy number. A case study to find the priority order of four destinations d1,d2,d3,d4 along with the maximum dynamic flow determining is provided to illustrate the proposed method. The hesitant fuzzy TOPSIS evaluated the priority order of destinations for evacuation the maximum number of aggrieved as {d4,d3,d1,d2} along with maximum lexicographic flow with partial lane reversal 62˜ flow units. The method for finding the spreads without blurring a fuzzy number was used to determine the final triangular number (50, 62, 75) units.The sensitivity analysis shows that in all ten cases, the alternatives have unique rank: A4 is the best, A2 is the worst. In the pessimistic case, DMs prefers the alternative that is furthest to FNIS. In the optimistic case, DMs prefer the alternative that is closest to FPIS. We proved that the proposed algorithm is reliable because despite the changes in parameters ω, the same priority order of alternatives is chosen. The obtained results are confirmed by efficiency frontier and coincide with the results of original TOPSIS, Doukas et al.’s, method, Kuo’s method, Yoon and Kim’s behavioral TOPSIS.

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.

Numerical study of the interaction between an internal solitary wave and a submerged extended cylinder using OpenFOAM

To investigate the impacts of cylinder extended length, rotation angle and wave incidence angle on hydrodynamic characteristics, the numerical simulation of an internal solitary wave (ISW) interaction with a submerged extended cylinder is conducted by using OpenFOAM. The accordance between the experimental data and numerical simulation forces on the isolated cylinder shows that the OpenFOAM-based numerical model can deal with the ISW-submerged cylinder interaction. The distributions of forces, maximum dynamic pressure forces and flow field around the submerged extended cylinder are discussed to reveal the hydrodynamic characteristics of submerged cylinder under the ISW action. The horizontal forces on an extended cylinder are found to change from increasing to decreasing with the increase in extended length when the extended length is 2.5 times the cylinder diameter. The forces on the cylinder are sensitive to the change in the rotation angle. Both the horizontal loads and the evolution of the wave profile are easily affected by the change in incidence angle. The vortices shedding around the cylinder varies with the change of wave incidence angle. Investigation of the ISW and a submerged cylinder interactions can help to understand the mechanism of structural damage in internal wave ocean environment.

FlowLoc problems with maximum excess flow

In order to reduce huge losses from natural and human caused disasters, an appropriate strategy of evacuation planning, preparedness, response and recovery is very essential. The placement of facilities in appropriate arcs of a network plays an important role for the optimal evacuation planning. Such an efficient assignment should respect the least decreases on the arc capacity and hence should minimize the reduction of the optimal flow value. On the other hand, due to the bottleneck capacity, not all flows out from the sources may reach to the sinks. To maximize the flow value, the resulting excess flow can be stored at intermediate nodes that are comparatively safer than the sources.By integrating both concepts, we introduce the single-source single-sink maximum static and dynamic flow location (FlowLoc) problems with excess flow storage allowed. Our objective here is to minimize the reduction of maximum flow in an evacuation network where given facilities are placed on arcs and excess flow is stored at intermediate nodes. We provide their mathematical models and also present polynomial time algorithms to solve them. For the maximum static and dynamic multi-FlowLoc problems where multiple facilities are located, we give their mathematical models and present polynomial time heuristics. These solutions here are practically important, especially in evacuation planning, to maximize the flow value out of the disastrous places by locating the facilities.

Abstract network flow with intermediate storage for evacuation planning

The abstract network is a generalization of the classical network which is associated with the set of elements and linearly ordered subset of elements, known as paths, that satisfy switching property. Each element is equipped with an integral capacity of two types, movement capacity which transships the flow from the element and storage capacity which stores the flow at the element. In the evacuation process, every individual may not be sent to the final destination due to capacity or time constraints. So, the storage of evacuees at intermediate elements can be a milestone to save the life of people. In this paper, we present polynomial time algorithms to solve maximum static, lexicographic maximum static and maximum dynamic flow problems in an abstract network. We also present the solution procedure to these problems in the abstract contraflow networks having symmetric as well as asymmetric transit times between the pair of adjacent elements.

Dynamic Multicommodity Contraflow Problem with Asymmetric Transit Times

A maximum dynamic multicommodity flow problem concerns with the transportation of several different commodities through the specific source-sink path of an underlying capacity network with the objective of maximizing the sum of commodity flows within a given time horizon. Motivated by the uneven road condition of transportation network topology, we introduce the dynamic multicommodity contraflow problem with asymmetric transit times on arcs that increase the outbound lane capacities by reverting the orientation of lanes towards the demand nodes. Moreover, a pseudo-polynomial time algorithm by using a time-expanded graph and an FPTAS by using aΔ-condensed time-expanded network are presented.

Maximum Multicommodity Flow with Intermediate Storage

The multicommodity flow problem deals with the transshipment of more than one commodity from respective sources to corresponding sinks without violating the capacity constraints. Due to the capacity constraints, flows out from the sources may not reach their sinks, and so, the storage of excess flows at intermediate nodes plays an important role in the maximization of flow values. In this paper, we introduce the maximum static as well as maximum dynamic multicommodity flow problems with intermediate storage. We present polynomial and pseudopolynomial time algorithms for the former and latter problems, respectively. We also present the solution procedures to these problems in contraflow network having symmetric as well as asymmetric arc transit times. We transform the solutions in continuous-time settings by using natural transformation.

Network Flow with Intermediate Storage: Models and Algorithms

Various network flow models, such as a flow maximization, a time minimization, a cost minimization, or a combination of them, have already been investigated. In most of the cases, they are considered subject to the flow conservation constraints. Here, we investigate the network flow models with intermediate storage, i.e., the inflow may be greater than the outflow at intermediate nodes. We introduce a maximum static and a maximum dynamic flow problem where an intermediate storage is allowed. Then, polynomial time algorithms are presented to solve these problems in two terminal general networks. We also study the earliest arrival property of the maximum dynamic flow in two terminal series-parallel networks and present its efficient solution procedure with intermediate storage. Moreover, we introduce a dynamic contraflow model with intermediate storage and present a polynomial time algorithm to solve the maximum dynamic contraflow problem in two terminal networks. We also solve an earliest arrival contraflow problem with intermediate storage. Our investigation is focused to solve the evacuation planning problem where the intermediate storage is permitted.

Efficient Algorithms on Multicommodity Flow over Time Problems with Partial Lane Reversals

The multicommodity flow problem arises when several different commodities are transshipped from specific supply nodes to the corresponding demand nodes through the arcs of an underlying capacity network. The maximum flow over time problem concerns to maximize the sum of commodity flows in a given time horizon. It becomes the earliest arrival flow problem if it maximizes the flow at each time step. The earliest arrival transshipment problem is the one that satisfies specified supplies and demands. These flow over time problems are computationally hard. By reverting the orientation of lanes towards the demand nodes, the outbound lane capacities can be increased. We introduce a partial lane reversal approach in the class of multicommodity flow problems. Moreover, a polynomial-time algorithm for the maximum static flow problem and pseudopolynomial algorithms for the earliest arrival transshipment and maximum dynamic flow problems are presented. Also, an approximation solution to the latter problem is obtained in polynomial-time.

Numerical investigation of an internal solitary wave interaction with horizontal cylinders

An internal solitary wave (ISW) interaction with different type horizontal cylinders are investigated. The forces induced by ISW on horizontal cylinders are presented. Distributions of the maximum dynamic pressures and flow fields around cylinders are described. Good agreements between numerical results and experimental data of forces on the isolated cylinder are achieved. For the extended cylinder, the vertical forces per unit length increase with the increase in the extended length. The horizontal forces on the extended cylinder change from increasing to decreasing after the centre-to-centre distance exceeding 3.5 times the diameter of the isolated cylinder. The peak and valley values of maximum dynamic pressures impose on the lower side of the leading edge and upper side of the trailing edge of the extended cylinder, respectively. The horizontal forces on each of the two tandem cylinders with different centre-to-centre distances are smaller than those on the isolated cylinder, indicating that the two cylinders play a role to protect each other. The comparison of forces between the two cylinders indicates that larger horizontal forces and smaller vertical forces are imposed on the rear cylinder. The maximum dynamic pressures around the rear cylinder are smaller than those around the front cylinder.

Minimum Clearance Time on the Prioritized Integrated Evacuation Network

The evacuation planning problem can be viewed as different variants of dynamic flow maximization and time minimization problems. An optimal solution to the latter problem sends a given amount of flow from disaster zones to safe zones in minimum time. We solve this problem on an embedded integrated network of a prioritized primary and a bus-routed secondary sub-networks. For a lexicographically maximum (lex-max) dynamic flow problem, we are given a time horizon and a prioritized network, where we need a feasible dynamic flow that lexicographically maximizes the flow amount leaving each terminal respecting the priority. Here, we use the quickest transshipment partial arc reversal strategy to collect the evacuees in minimum time from the disaster zones to the pickup locations of the primary sub-network. By treating such pickup locations as sources, the available set of transit-buses is assigned in the secondary sub-network to shift the evacuees finally to the sinks on the first-come-first-serve basis. This novel approach proposed here is better suited for the simultaneous flow of evacuees with minimum waiting delay at such pickup locations in the integrated evacuation network topology. The lane reversal strategy significantly reduces the evacuation time, whereas reversing them only partially has an additional benefit that the unused road capacities can be used for supplying emergency logistics and allocating facilities as well.

Lexicographically Maximum Dynamic Flow with Vertex Capacities

We consider an evacuation planning problem in the sense of computing a feasible dynamic flow lexicographically maximizing the amount of flow entering a set of terminals with respect to a given prioritization and given vertex capacities. We propose a polynomial time algorithm for the static version of the problem and a pseudo-polynomial time algorithm for the dynamic case. We show that by neglecting the vertex capacities, the dynamic version can be solved in polynomial time by using temporally repeated flows.

Efficient Dynamic Flow Algorithms for Evacuation Planning Problems with Partial Lane Reversal

Contraflow technique has gained a considerable focus in evacuation planning research over the past several years. In this work, we design efficient algorithms to solve the maximum, lex-maximum, earliest arrival, and quickest dynamic flow problems having constant attributes and their generalizations with partial contraflow reconfiguration in the context of evacuation planning. The partial static contraflow problems, that are foundations to the dynamic flows, are also studied. Moreover, the contraflow model with inflow-dependent transit time on arcs is introduced. A strongly polynomial time algorithm to compute approximate solution of the quickest partial contraflow problem on two terminal networks is presented, which is substantiated by numerical computations considering Kathmandu road network as an evacuation network. Our results show that the quickest time to evacuate a flow of value 100,000 units is reduced by more than 42% using the partial contraflow technique, and the difference is more with the increase in the flow value. Moreover, the technique keeps the record of the portions of the road network not used by the evacuees.

Non-conserving Flow Aspect of Maximum Dynamic Flow Problem

Shifting as many people as possible from disastrous area to safer area in a minimum time period in an efficient way is an evacuation planning problem (EPP). Modeling the evacuation scenarios reflecting the real world characteristics and investigation of an efficient solution to them have become a crucial due to rapidly increasing number of natural as well as human created disasters. EPPs modeled on network have been extensively studied and the various efficient solution procedures have been established where the flow function satisfies the flow conservation at each intermediate node. Besides this, the network flow problem in which flow may not be conserved at nodes necessarily could also be useful to model the evacuation planning problem. This paper proposes an efficient solution procedure for maximum flow evacuation planning problem of later kind on a single-source-single-sink dynamic network with integral arc capacities with holding capability of flow (evacuees) in the temporary shelter at intermediate nodes. Journal of the Institute of Engineering, 2018, 14(1): 107-114

The Maximum Flows in Planar Dynamic Networks

An nontrivial extension of the maximal static flow problem is the maximal dynamic flow model, where the transit time to traverse an arc is taken into consideration. If the network parameters as capacities, arc traversal times, and so on, are constant over time, then a dynamic flow problem is said to be stationary. Research on flow in planar static network is motivated by the fact that more efficient algorithms can be developed by exploiting the planar structure of the graph. This article states and solves the maximum flow in directed (1, n) planar dynamic networks in the stationary case.

Atomic routing in a deterministic queuing model

The issue of selfish routing through a network has received a lot of attention in recent years. We study an atomic dynamic routing scenario, where players allocate resources with load dependent costs only for some limited time.Our paper introduces a natural discrete version of the deterministic queuing model introduced by Koch and Skutella (2011). In this model the time a user needs to traverse an edge e is given by a constant travel time and the waiting time in a queue at the end of e. At each discrete time step the first ue users of the queue proceed to the end vertex of e, where ue denotes the capacity of the edge e. An important aspect of this model is that it ensures the FIFO property.We study the complexity of central algorithmic questions for this model such as determining an optimal flow in an empty network, an optimal path in a congested network or a maximum dynamic flow and the question whether a given flow is a Nash equilibrium.For the bottleneck case, where the cost of each user is the travel time of the slowest edge on her path, the main results here are mostly bad news. Computing social optima and Nash equilibria turns out to be NP-complete and the Price of Anarchy is given by the number of users.We also consider the makespan objective (arrival time of the last user) and show that optimal solutions and Nash equilibria in these games, where every user selfishly tries to minimize her travel time, can be found efficiently.

Maximum dynamic network flow interdiction problem: New formulation and solution procedures

We consider the dynamic version of the maximum flow network interdiction problem; that is, we assume a positive number is assigned to each arc which indicates the traversal time of the flow through that arc. We also assume that an intruder uses a single resource with limited budget to interrupt the flow of a single commodity through the network within a given time limit of T. A new formulation based on the concept of Temporally Repeated Flow (TRF) is presented. The problem is then solved using Benders' decomposition. Another solution method, based on the most vital arcs in a network is also discussed. Finally, some computational results are reported.

The inverse maximum dynamic flow problem

We consider the inverse maximum dynamic flow (IMDF) problem. IMDF problem can be described as: how to change the capacity vector of a dynamic network as little as possible so that a given feasible dynamic flow becomes a maximum dynamic flow. After discussing some characteristics of this problem, it is converted to a constrained minimum dynamic cut problem. Then an efficient algorithm which uses two maximum dynamic flow algorithms is proposed to solve the problem.