Dynamic flows with supply and demand

We are given a network G = (N, A, h, c) with node set N, arch set A, time function h, capacity function c, and P the set of periods, s the source and s′ the sink of the network G. Associated with s, there is a non-negative real number q(t) called the supply of source s at time t, and with s′ – a nonnegative real number q′(t) called the demand of sink s′ at time t, t is an element of P. The objective is to determine the existence of a dynamic flow in G for p periods, so that the demands at sink s′ can be fulfilled from the supplies at the source s. A numerical example is presented.