Max-flow min-cut theorem for directed fuzzy incidence networks

Modern networks like the internet and data grids are rapidly evolving and quickly growing in size. The investigation and study of their dynamism and stability necessitate the use of graph-theoretic approaches. Fuzzification of the network is unavoidable in order to incorporate their vast size. This paper proposes a network model, namely directed fuzzy incidence network (DFIN), suitable for dealing with networks of extraneous flows and support. It is a special kind of directed fuzzy incidence graphs that are both node and edge capacitated. Transportation networks may be most successfully assessed when they are regarded as directed fuzzy incidence networks. This study also concentrates on legal flow, saturated and unsaturated arc, arc cut, and legal flow enhancing path in a directed fuzzy incidence network. Legal flow on a directed fuzzy incidence network is defined as a function on the arc set in such a way that the legal flow of an arc never exceeds the corresponding legal incidence strength of the arc. Moreover, it holds node capacity constraint as well as flow conservation constraint. It is shown that the resultant legal flow out of the source is equal to the resultant legal flow into the sink for any legal flow defined on the directed fuzzy incidence network. Also, it is proved that the value of a legal flow can be determined using any arc cut in the directed fuzzy incidence network. The relation between the value of a legal flow and any arc cut in the directed fuzzy incidence network is established and a necessary and sufficient condition for their equality is obtained. Moreover, an equivalent condition for a legal flow to become maximum is given using enhancing paths. The ultimate goal of the work is to provide a DFIN-analog of max-flow min-cut theorem in graph theory. In addition, the paper proposes an algorithm for determining maximum legal flow in a directed fuzzy incidence network and illustrates it with an application in transport of commodities.