# Consider the problem of finding the maximum flow through a network from a set of supply points to…

Consider the problem of finding the maximum flow through a network from a set of supply points to a set of demand points. If there are alternate optima (different ways of maximizing the flow), you want to find the way that minimizes the total shipment cost. Associated with each link in the network is: a lower bound, lij; and upper bound, uij , and a unit cost, cij . An example network follows.

The problem is characterized by the inputs shown on the following page. Nodes 51 , 52, and 53 are supply points; nodes D9 and D1Q are demand points. Nodes 4 through 8 are transshipment points. Note that no flow is permitted between nodes 51 and 5 or between nodes 5 3 and 4. However, flow is permitted between nodes 51 and 6 and between nodes 53 and 8.

(a) Show how this problem tha t of maximizing the flow from the source nodes to the destination nodes at minimum total cost ca n be structured as an out-of-kilter flow problem. That is, draw the network that would be used in an out-of-kilter flow problem, clearly identifying the lower bounds, upper bounds, and unit costs on all links in the original network (such as the one shown above). If you elect to add any nodes or links, clearly label all link costs associated with these links as well.

(b) Suppose you want to find the minimum cost way of shipping as much as possible through the network subject to the condition that no shipment costs more than C m a x to ship. (That is, all shipments must cost C m a x or less to ship.) Discuss how this can be formulated as an out of-kilter-flow problem. Be careful here! You may need to change some of the costs in the original network to be sure you get the right answer!

(c) For the network shown above, find the minimum cost way of shipping the maximum flow from the supply nodes to the demand nodes using the MENU-OKF algorithm.

Hint: Before typing the inputs into MENU-OKF, read part (d). Careful specification of the inputs will allow you to solve both problems with relatively few changes to the network.

(d) For the network shown above, find the minimum cost way of shipping the maximum possible flow subject to the additional condition that no shipment costs more than 25 units.

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