Network Flow Programming - Example Problem - Power Distribution
 Consider a regional power system with three generating stations: A, B, and C. Each station serves its own local area. Three outlying areas are also served by the system: X, Y, and Z. The power demanded at areas X, Y, and Z is 25 MW (megawatts), 50 MW, and 30 MW, respectively. The maximum generating capacity beyond local requirements and the cost of generation at the three stations is shown in the table. Power can be transmitted between any pair of generating stations, but 5% of the amount is lost. Power can be transmitted from some of the generating stations to the outlying areas, but 10% of the amount is lost. Lines exist from stations A and C to X, from B and C to Y, and from A and B to Z. Our goal is find the minimum cost power distribution plan between generating stations and to outlying areas. The network model for this problem consists of nodes and arcs as shown in the figure. The nodes are the circles and represent the generating stations and cities. The arcs are the directed line segments between nodes. They represent the transmission lines between generating stations and/or cities. Network Flow Model of Power Distribution Problem The numbers adjacent to the nodes in the square brackets represent flows entering the network or flows leaving the network, positive numbers for flows entering and negative numbers for flows leaving. Power enters at the generation stations, nodes A, B, and C through arcs that enter these nodes. Power leaves the network at nodes X, Y, and Z, where the negative numbers at the nodes indicates the amounts to be withdrawn at the nodes. The arcs for this case have three parameters, the upper bound, cost and gain. The upper bound for each transmission arc is M, indicating a large number. For an arc that passes from node i to node j, the gain multiplies the flow leaving node i to obtain the flow that enters node j. For this problem the gain factors represent the losses of power in the transmission lines. A solution to the network model is an assignment of flows to the arcs that satisfies the flow requirements at the nodes. Flow is conserved at each node in that the total flow entering must equal the total flow leaving. An arc that touches only one node, such as those entering A, B and C, contributes only to the conservation equation for that node. The optimum solution minimizes cost.

Operations Research Models and Methods
Internet
by Paul A. Jensen