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The following paper describes the method of the maximum flow finding in fuzzy dynamic graph with the given vitality degree. The peculiarity of the problem is in fuzzy nature of the considered network and given vitality parameters assigned to arcs of the graph. Network's parameters, such as arc capacities and vitality degrees can vary over time, because they have transit character. Such methods can be applied in the real networks solving the task of the optimal cargo transportation. The flow tasks arising in the study of transportation networks are relevant due to their wide practical application. Conventional flow tasks allow researchers to find the maximum amount of traffic between selected nodes on the road map, determine the routes of the optimal cost. Dynamic networks describe complex systems, which parameters can vary over time. Algorithms underlying these tasks allow solving optimization problems taking into account the flow departure time, transit times along the arcs of the graph. However, the flow tasks and methods of its solution are presented in the references in crisp conditions, while changes in environment, variations in petrol prices, measurement errors influence such network parameters, as arc capacities and transmission costs. Therefore, these tasks should be considered in the fuzzy conditions and we turn to the fuzzy graphs for solving such a problems. 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. Vitality parameter peculiar to arcs of the network usually isn't taken into account while studying networks. Nowadays, the vitality of the network is studied less, while railways and roads include the complex objects, such as stations, distillation ways, culverts, wagon, passenger and cargo managements. Thus, conventional definition of "vitality" was introduced by the authors H. Frank and I. Frisch in (2) as sensitivity of the network to damages. However, vitality applied to the networks is ability of its objects and links among them to be resistant to weather conditions, traffic accidents and its combinations, and save and restore (fully or partially) objects themselves and their connections, arc capacities of the network's sections in case of damage. Sometimes network's parameters can be set qualitatively. Thus, one can set the notion "vitality degree" considering the roads and railways. In this case "vitality degree" is considered as probability of trouble-free operation of the road section and some subjective value, such as importance and reliability, etc. Therefore, the method of the maximum flow finding in fuzzy dynamic graph with the given vitality degree is proposed in the present paper. The paper is structured as follows. In the Section 2 we give literature review of the flow and vitality problems in networks. Section 3 contains basic concepts and definitions. Section 4 presents the proposed method of the maximum flow finding in fuzzy dynamic graph with the given vitality degree. Section 5 provides numerical example illustrating the main steps of the proposed method.

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