Multi-commodity Maximum Flow Determining in a Fuzzy Graph with Vitality Degrees

Multi-commodity flow finding is an important task of transportation planning and optimization. This paper describes one of its variations - the task of the two-commodity flow obtaining in a fuzzy graph considering vitality degree. The core of this task is in fuzzy arc parameters, such as arc capacities and degrees of vitality. Vitality parameter in these terms can be viewed as probability of trouble-free operation of the road section and some subjective value, such as importance and reliability. The proposed method utilizes modified augmenting path algorithm that allows considering fuzzy nature of the arc parameters and vitality degree assigned to each arc. The following method can be exploited when we plan transportation of two types of commodities, each with their own source and sink, for example, passenger and cargo trains. Numerical examples of the proposed task are given.