scholarly journals A Simple Algorithm for Finding Maximal Network Flows and an Application to the Hitchcock Problem

1957 ◽  
Vol 9 ◽  
pp. 210-218 ◽  
Author(s):  
L. R. Ford ◽  
D. R. Fulkerson
Keyword(s):  
Network Flow ◽  
Network Flows ◽  
Simple Algorithm ◽  
Rand Corporation ◽  
Network Flow Problem ◽  
Intermediate Node ◽  
Flow Capacity ◽  
Source To Sink ◽  
Image Position ◽  
Source Sink

The network-flow problem, originally posed by T. Harris of the Rand Corporation, has been discussed from various viewpoints in (1; 2; 7; 16). The problem arises naturally in the study of transportation networks; it may be stated in the following way. One is given a network of directed arcs and nodes with two distinguished nodes, called source and sink, respectively. All other nodes are called intermediate. Each directed arc in the network has associated with it a nonnegative integer, its flow capacity. Source arcs may be assumed to be directed away from the source, sink arcs into the sink. Subject to the conditions that the flow in an arc is in the direction of the arc and does not exceed its capacity, and that the total flow into any intermediate node is equal to the flow out of it, it is desired to find a maximal flow from source to sink in the network, i.e., a flow which maximizes the sum of the flows in source (or sink) arcs.Thus, if we let P1 be the source, Pn the sink, we are required to find xij (i,j =1, . . . , w) which maximize

scholarly journals Minimum cost network flows: Problems, algorithms, and software

2013 ◽  
Vol 23 (1) ◽  
pp. 3-17 ◽  
Author(s):  
Angelo Sifaleras
Keyword(s):  
Network Flow ◽  
Flow Problem ◽  
Network Flows ◽  
Minimum Cost ◽  
Network Flow Problem ◽  
Software Packages ◽  
Wide Range ◽  
Solution Methods ◽  
Minimum Cost Network Flow ◽  
Algorithmic Approaches

We present a wide range of problems concerning minimum cost network flows, and give an overview of the classic linear single-commodity Minimum Cost Network Flow Problem (MCNFP) and some other closely related problems, either tractable or intractable. We also discuss state-of-the-art algorithmic approaches and recent advances in the solution methods for the MCNFP. Finally, optimization software packages for the MCNFP are presented.

scholarly journals Efficient Solution Approach to Maximum Flow Evacuation Planning Problem without Flow Conservation Aspect

2018 ◽  
Vol 13 (1) ◽  
pp. 108-116
Author(s):  
Phanindra Prasad Bhandari ◽  
Shree Ram Khadka
Keyword(s):  
Network Flow ◽  
Efficient Solution ◽  
Maximum Flow ◽  
Evacuation Planning ◽  
Planning Problem ◽  
Intermediate Node ◽  
Solution Approach ◽  
Solution Strategies ◽  
Source Sink ◽  
Flow Conservation

 An attempt of shifting as more people as possible and/or their logistics from a dangerous place to a safer place is an evacuation planning problem. Such problems modeled on network have been extensively studied and the various efficient solution procedures have been established. The solution strategies for these problems are based on source-sink path augmentation and 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 node necessarily could also be used to model the evacuation planning problem. This paper proposes a model for maximum flow evacuation planning problem on a single-source-single-sink static network with integral arc capacities with holding capability of evacuees in the temporary shelter at intermediate nodes and extends the model into the dynamic case. Journal of the Institute of Engineering, 2017, 13(1): 108-116

scholarly journals Knights Exchange Puzzle—Teaching the Efficiency of Modeling

2020 ◽  
Author(s):  
Mehdi Iranpoor
Keyword(s):  
Network Flow ◽  
Network Flows ◽  
Optimization Problems ◽  
Minimum Cost ◽  
Network Flow Problem ◽  
Binary Programming ◽  
Combinatorial Optimization Problems ◽  
Minimum Cost Network Flow ◽  
Binary Formulation

Puzzles and games enhance the quality of teaching by creating an enjoyable, interactive, and playful atmosphere. The knight exchange is a famous, very old, and amusing game on the chessboard. This puzzle was used by the author to teach modeling in a mathematical programming course designed for graduate students. The aim was to teach the students the efficiency of the models. Accordingly, first, a binary programming formulation was developed. This formulation was, however, found to be inefficient, and tremendous time (i.e., more than four hours) and a large amount of processing memory were needed to solve the puzzle. The puzzle was subsequently formulated as a minimum cost network flow problem. The latter formulation outperformed the general binary formulation by solving the puzzle in less than a minute. The network formulation could also save the required processing memory. The results could help students to learn the value of modeling combinatorial optimization problems as network flows.

Review of Network Flow Algorithms David P. Williamson

2021 ◽  
Vol 52 (1) ◽  
pp. 12-15
Author(s):  
S.V. Nagaraj
Keyword(s):  
Graduate Students ◽  
Real World ◽  
Network Flow ◽  
Network Flows ◽  
Optimization Problems ◽  
Capacity Constraints ◽  
Incoming Flow ◽  
Network Flow Problems ◽  
Flow Problems ◽  
Real World Problems

This book is on algorithms for network flows. Network flow problems are optimization problems where given a flow network, the aim is to construct a flow that respects the capacity constraints of the edges of the network, so that incoming flow equals the outgoing flow for all vertices of the network except designated vertices known as the source and the sink. Network flow algorithms solve many real-world problems. This book is intended to serve graduate students and as a reference. The book is also available in eBook (ISBN 9781316952894/US$ 32.00), and hardback (ISBN 9781107185890/US$99.99) formats. The book has a companion web site www.networkflowalgs.com where a pre-publication version of the book can be downloaded gratis.

Bilevel Model for Adaptive Network Flow Problem

2018 ◽  
Vol 64 ◽  
pp. 105-114 ◽  
Author(s):  
Jean-François Baffier ◽  
Pierre-Louis Poirion ◽  
Vorapong Suppakitpaisarn
Keyword(s):  
Network Flow ◽  
Flow Problem ◽  
Adaptive Network ◽  
Network Flow Problem
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