Method for the minimum cost maximum flow determining in fuzzy dynamic network with nonzero flow bounds

Abstract

In this paper the method for the minimum cost flow determining in fuzzy dynamic network with nonzero lower flow bounds is proposed. Considered method takes into account fuzzy nature of networks' parameters and deals with fuzzy arc flow bounds, fuzzy transmission costs. Fuzzy arc flow bounds, costs and transit times can vary depending on the flow departure time. The method allows introducing the notion of "fuzzy network" instead of "stationary-dynamic" network, considered in conventional flow literature. The sphere of application for considered problem is presented.Keywords: Fuzzy dynamic network, fuzzy minimum cost flow with nonzero lower flow bounds.

Maximum Flow Finding in Fuzzy Dynamic Graph with the Given Vitality Degree

“…Dynamic nature of the network's parameters and their opportunity to change in time isn't taken into account in conventional sources that is incorrect in general. The authors suggest calling these networks "stationary-dynamic" and introducing the notion of dynamic networks in [1].…”

“…Fuzzy stationary-dynamic network [1] is a network in the form of the fuzzy directed graph ( , )  is set to each arc of the graph. There is time horizon {0,1,..., } Tp  , which determines, that all flow units sent from the source must arrive at the sink not later, than in the time p. ij  is a positive number.…”

“…Dynamic tasks considered in the conventional flow literature take into account transit times along the arcs of the graph, while network's parameters are constant. We suggest to consider dependence of arc capacities and transmission costs on flow departure time and operate with fully dynamic networks instead of stationary-dynamic ones [1], using the notions of the time-expanded graphs.…”