Abstract

The maximum flow problem is also one of the highly regarded problems in the field of optimization theory in which the objective is to find a feasible flow through a flow network that obtains the maximum possible flow rate from source to sink. The literature demonstrates that different techniques have been developed in the past to handle the maximum amount of flow that the network can handle. The Ford-Fulkerson algorithm and Dinic's Algorithm are the two major algorithms for solving these types of problems. Also, the Max-Flow Min-Cut Theorem, the Scaling Algorithm, and the Push–relabel maximum flow algorithm are the most acceptable methods for finding the maximum flows in a flow network. In this novel approach, the paper develops an alternative method of finding the maximum flow between the source and target nodes of a network based on the "max-flow." Also, a new algorithmic approach to solving the transportation problem (minimizing the transportation cost) is based upon the new maximum flow algorithm. It is also to be noticed that this method requires a minimum number of iterations to achieve optimality. This study's algorithmic approach is less complicated than the well-known meta-heuristic algorithms in the literature. 

Highlights

  • The flow network is a directed graph where V is an n-set of nodes that get a flow and E is an m-set of directed edges that have a capacity

  • In the previous few years, different types of algorithms for solving this problem have been proposed, for example, the network unraveling technique based on the Ford-Fulkerson max-flow algorithm (1956), A Parallel Ford-Fulkerson Algorithm For Maximum Flow Problem (Harris and Ross, 1956), Maximum Flow Problem (Ahuja, 1989; Karzanova, 1974)

  • This paper presents some new strategies for taking care of the maximal network flow problem with applications to solving the transportation problem

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Summary

Introduction

The flow network is a directed graph where V is an n-set of nodes that get a flow and E is an m-set of directed edges that have a capacity. In the previous few years, different types of algorithms for solving this problem have been proposed, for example, the network unraveling technique based on the Ford-Fulkerson max-flow algorithm (1956), A Parallel Ford-Fulkerson Algorithm For Maximum Flow Problem (Harris and Ross, 1956), Maximum Flow Problem (Ahuja, 1989; Karzanova, 1974). Transportation problems have been generally considered in Operation Research It is one of the fundamental problems of the network flow problem (Dinice, 1970; Elias et al, 1956; Fulkerson & Dantzig, 1955), which is normally used to minimize the transportation cost for ventures with a couple of sources to a couple of objectives. We present a few new effective algorithms for the summarized maximum flow problem

The Key Definitions
Predecessor Cost Matrix for Max Flow
Illustration of the Proposed Algorithm
New Algorithm
New Algorithm for TP
Comparative Assessment with Max Flow
Comparative Assessment with TP
Conclusion
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