Closed parasitic flow loops and dominated loops in networks

The paper raises awareness of the presence of closed parasitic flow loops in the solutions of published algorithm for maximising the throughput flow in networks. If the rooted commodity is interchangeable commodity, a closed parasitic loop can effectively be present even if the routed commodity does not physically travel along a closed loop. The closed parasitic flow loops are highly undesirable loops of flow, which effectively never leave the network. Parasitic flow loops increase the cost of transportation of the flow unnecessarily, consume residual capacity from the edges of the network, increase the likelihood of deterioration of perishable products, increase congestion and energy wastage. Accordingly, the paper presents a theoretical framework related to parasitic flow loops in networks. By using the presented framework, it is demonstrated that the probability of existence of closed and dominated flow loops in networks is surprisingly high. The paper also demonstrates that the successive shortest path strategy for minimising the total length of transportation routes from multiple interchangeable origins to multiple destinations fails to minimise the total length of the routes. It is demonstrated that even in a network with multiple origins and a single destination, the successive shortest path strategy could still fail to minimise the total length of the routes. By using the developed theoretical framework, it is shown that a minimum total length of the transportation routes in a network with multiple interchangeable origins, is attained if and only if no closed parasitic flow loops and dominated flow loops exist in the network. Accordingly, an algorithm for minimising the total length of the transportation routes by eliminating all dominated parasitic flow loops is proposed