Transportation on Networks

2016 ◽  
Author(s):  
Alfred Galichon
Keyword(s):  
Shortest Path ◽  
Assignment Problem ◽  
Network Flow ◽  
Flow Problem ◽  
Optimal Assignment ◽  
Shortest Path Problems ◽  
Flow Problems ◽  
Optimal Flow ◽  
Optimal Network ◽  
The Difference

This chapter considers the optimal network flow problem, which is a generalization of the optimal assignment problem considered in Chapter 3. In optimal flow problems, one considers a network of cities, or edges, to move a distribution of mass on supply nodes to a distribution of mass on demand nodes. The difference from a standard optimal assignment problem is that the matching surplus associated with moving from a supply location to a demand location is not necessarily directly defined; instead, there are several paths from the supply location to the demand location, among these some yield maximal surplus. Therefore, both the optimal assignment problem and the shortest path problem are instances of the optimal flow problem; these instances are representative in the sense that any optimal flow problem may be decomposed into an assignment problem and a number of shortest path problems. The chapter shows how to easily compute these problems using linear programming.

Networks Flow Applications

2013 ◽  
pp. 246-256
Author(s):  
Alireza Boloori ◽  
Monirehalsadat Mahmoudi
Keyword(s):  
Network Flow ◽  
Large Scale ◽  
Flow Problem ◽  
Minimum Cost ◽  
Maximum Flow ◽  
Shortest Path Problems ◽  
Flow Problems ◽  
Large Scale Integration ◽  
Minimum Cost Flow Problem ◽  
Scale Integration

In this chapter, some applications of network flow problems are addressed based on each type of problem being discussed. For example, in the case of shortest path problems, their concept in facility layout, facility location, robotics, transportation, and very large-scale integration areas are pointed out in the first section. Furthermore, the second section deals with the implementation of the maximum flow problem in image segmentation, transportation, web communities, and wireless networks and telecommunication areas. Moreover, in the third section, the minimum-cost flow problem is discussed in fleeting and routing problems, petroleum, and scheduling areas. Meanwhile, a brief explanation about each application as well as some corresponding literature and research papers are presented in each section. In addition, based on available literature in each of these areas, some research gaps are identified, and future trends as well as chapter’s conclusion are pointed out in the fourth section.

scholarly journals A combinatorial arc tolerance analysis for network flow problems

2005 ◽  
Vol 2005 (2) ◽  
pp. 83-94 ◽  
Author(s):  
P. T. Sokkalingam ◽  
Prabha Sharma
Keyword(s):  
Cost Function ◽  
Network Flow ◽  
Flow Problem ◽  
Tolerance Analysis ◽  
Network Flow Problems ◽  
Total Cost ◽  
Flow Problems ◽  
Optimal Flow ◽  
The Given ◽  
Total Cost Function

For the separable convex cost flow problem, we consider the problem of determining tolerance set for each arc cost function. For a given optimal flow x, a valid perturbation of cij(x) is a convex function that can be substituted for cij(x) in the total cost function without violating the optimality of x. Tolerance set for an arc(i,j) is the collection of all valid perturbations of cij(x). We characterize the tolerance set for each arc(i,j) in terms of nonsingleton shortest distances between nodes i and j. We also give an efficient algorithm to compute the nonsingleton shortest distances between all pairs of nodes in O(n3) time where n denotes the number of nodes in the given graph.

scholarly journals Random assignment and shortest path problems

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Johan Wästlund
Keyword(s):  
Limit Theorem ◽  
Shortest Path ◽  
Complete Graph ◽  
Assignment Problem ◽  
Shortest Path Problem ◽  
Random Assignment ◽  
Shortest Path Problems ◽  
Parisi Formula ◽  
International Audience ◽  
Random Assignment Problem

International audience We explore a similarity between the $n$ by $n$ random assignment problem and the random shortest path problem on the complete graph on $n+1$ vertices. This similarity is a consequence of the proof of the Parisi formula for the assignment problem given by C. Nair, B. Prabhakar and M. Sharma in 2003. We give direct proofs of the analogs for the shortest path problem of some results established by D. Aldous in connection with his $\zeta (2)$ limit theorem for the assignment problem.

Arc tolerances in shortest path and network flow problems

Networks ◽  
1980 ◽  
Vol 10 (4) ◽  
pp. 277-291 ◽  
Author(s):  
D. R. Shier ◽  
Christoph Witzgall
Keyword(s):  
Shortest Path ◽  
Network Flow ◽  
Network Flow Problems ◽  
Flow Problems

scholarly journals Incremental Minimum Flow Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 1025
Author(s):  
Laura Ciupala ◽  
Adrian Deaconu
Keyword(s):  
Lower Bound ◽  
High Frequency ◽  
Network Flow ◽  
Flow Problem ◽  
Computation Method ◽  
Minimum Flow ◽  
Flow Problems ◽  
Initial Network ◽  
Minimum Flow Problem ◽  
Real World Problems

There are various situations in which real-world problems can be modeled and solved as minimum flow problems. Sometimes, in these situations, minor data changes may occur, leading to corresponding changes of the networks in which the practical problems are modeled as flow problems, such as slight variations in capacity or lower bound. For instance, the capacity or the lower bound of an arc may increase or decrease in time, leaving one with no other choice than finding the new minimum network flow. Given both the various ways in which the networks can be changed and the high frequency of these changes, it is desirable to find as fast a computation method for minimum flow as possible. This paper is focused on the cases that concern increasing and decreasing the capacity or the lower bound of an arc. For these cases, both the minimum flow algorithms and the dynamic minimum flow algorithms that are already known are inefficient. Our incremental algorithms for determining minimum flow in the modified network are more efficient than both the above-mentioned types of algorithms. The proposed method starts from the initial network minimum flow and solves the minimum flow problem in a significantly faster way than recalculating the new network minimum flow starting from scratch.

scholarly journals The Out-of-Kilter Algorithm and Its Applications to Network Flow Problems

2020 ◽  
pp. 76-97
Author(s):  
W. H. Moolman
Keyword(s):  
Network Flow ◽  
Network Flows ◽  
Optimal Solution ◽  
Minimum Cost ◽  
Pascal Program ◽  
Complementary Slackness ◽  
Initial Flow ◽  
Shortest Path Problems ◽  
Flow Problems ◽  
Minimum Cost Flow Problem

The out-of-kilter algorithm, which was published by D.R. Fulkerson [1], is an algorithm that computes the solution to the minimum-cost flow problem in a flow network. To begin, the algorithm starts with an initial flow along the arcs and a number for each of the nodes in the network. By making use of Complementary Slackness Optimality Conditions (CSOC) [2], the algorithm searches for out-of-kilter arcs (those that do not satisfy CSOC conditions). If none are found the algorithm is complete. For arcs that do not satisfy the CSOC theorem, the flow needs to be increased or decreased to bring them into kilter. The algorithm will look for a path that either increases or decreases the flow according to the need. This is done until all arcs are in-kilter, at which point the algorithm is complete. If no paths are found to improve the system then there is no feasible flow. The Out-of-Kilter algorithm is applied to find the optimal solution to any problem that involves network flows. This includes problems such as transportation, assignment and shortest path problems. Computer solutions using a Pascal program and Matlab are demonstrated.

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